гдз геометрия, билянин by Влад Микитюк

MOAvnb

I)) AM = x cm, BM = (r + 6) cnn. x'f (x ): 6) = 42; 2x -l 6 = 42; 2x = 36; r AM = la cw, BM = 18 + 6 = 24 (cur).

. CI4CTEMATU3AU|9 TA ySArAIbHEHH9 I METoAIB nnlxltvtsrpli

..,' onrng

$ 1 .1 . Ilpo

ll) AM = x cttr, BM = 2x cr,tt. x * 2x = 42i 3x = 42; x = 42 : 3; x = 14. AM =14clu, BM =2.L4 = 28 (cu). f) AM = 3x ctttt, BM = 4x c:lr. 3x-t 4x = 42;7x = 42; x ='6. AM = 6. 3 = 18 cltr, BM = 4 - 6 = 24 (cwt). A) AM = x cm, BM = 2x ctl. x t 2x = 42; 3x = 42; x = 42 : 3; t = 14. AM = 14 cw, BM = 2 . 74= 28 (cm).

norirry no6ygoey nnaxiruerpii

1.1.8, f. Ha npsMift ogna i rinrxa oAna Jrerr{r{rb mix 4eoua inruzuu (axc. II,). flpra qrolry Moxrnsi Tpt4 Br?rnagxrz BBaeMHoro poslrirqeuus AaHsx roqox Ha rlpg-

1.2,3 rprox roqox

uifi. 1) TouraA rerrtr,rrb milrc B i C, roAi aa arc. III, BC =AB + -l AC, are 1,38 4rvr + 2,72 aw + 1,34 gM. Touy rare poe-

-"""

uiuqenna HeMorrurrrBe.

2) Toura B .nesczrr uim A i C. AC = AB t BC, ane I,34 ptr + * 2,72 gM, + 1,38 ,qlr. Taxe posrrirqeHHs rorrox rarcorr{ He-

-"-*t-

MOXInBe.

3) Togi eAnne Mo*cJrr,rBe B3aeMHe poauirqenna rpbox roqor A, B i C raxe: Toqra C .nerrtr.rrr irdnc A i B. fleperiplrmo: AB =

1.9.3a axciouonIII,: IPOM = IPOA + ZAOM. A\ IAOM = xi ZPOA= x * 4O; rc + xc + 40 = 160; 2t + 40 = 16O; 2x = 120; x = 6O. IAOM = 6O'; IPOA. = 60' + 40' = 100'.

E)ZAOM=rciLPOA=x-60:

-r--1

AC + BC, 2,72 sNr = 1,38 ,qM -l 1,34 4ttt; 2,72 +nt = 2,72 arr. BiOnoailu: ro.{xa C JreJrcr{rb nri:x roqxauu A i B. =

r't x -

1.S.Ocri;rsxu

ea yuonoro

\2\/_

nnyrpiurnn rovrca ryra.AOB,

- mirx croponaula xyra AOB. OM npoxogurt Hexafi ZBOM = ,t,.ro+i IAOM = 2x. 3a aftc. III2 ZAOB = ZAOM + ZBOM,' ro

nporrnirr,

l5O = 2x

+.r,

150 =

.BOM = 50", .AOM Biinoaido: E) 100'.

3t, * = 50. .

cru.

-fi-h

1.8.

.--y4

=OA- OB = 7,3 - 5,8 = 1,5 (cu). _----.-- X rexarr xa ognift ninnparriri o no- 6----;---T rrarr(ciBoro ror{Horo o, ro sonra He poSAiJr,frorr,ca roqKoro O, ro6ro, ro.rxa O He Jrefirr,rrb rvrirx ro.*tarvru A i B. flrcu1o 6 ro.rxa A .nexa.na uint rorrr(aMLr B i O, ro BvxorryBaJracb 6ra pinnicrr OB = OA -l AB. A.te qe lreMorlcJrtlBo, 6o sa ymonoro OA > OB. Orxe, rovra B .nexrarr Minc ro.rxaMr,r,4 i O.

l:Iexafi ro.rrcr.r -4 , B, M nexanr na oAnift npsrvrift, ro4i rimxr oAHa s Hr,rx Jrerrclrrb ruix gnorta iurtrt4Ml4, i sa a&cioMoro nrzuiprona4na rigpiertin BrrrcoHyerbcs oAHa

cltt, BM = (x - 2\ crt. (x - 2) = 42:' 2x - 2 = 42; 2x = 44t = 22 cv, BM = 22 - 2 = 20 (cn).

fd*

B) ZAOD = rci ZAOM = O,6tci x * 0,6x= 160; 1,6* = 16O; x= 160 : 1,6; * = 100. /AOP = LOO'; ZAOM = 0,6 . 100" = 60'. f\ IAOM = tci ZPOA = 3xl x I 3r = 160; 4x = 160; r = 40. IAOM = 40'; ZPOA= 3 . 40" = 120". I) ZAOM = 3x; ZPOA= 5x; 3r 't 5x = 160; 8r = t$Q; a = 20. lAoM = 9..20" = 6o"; ZPOA = 5.20" = 100",

1.12, Iloscnesna gzr. N! 1.2.

624

= 110.

BiOnoeila: L,5 cm,

AM = x

x* AM

+ AB; AB Ocxinsnrz ro.rxr,r A i B

ctlc.

flpr.r posn'aeanni 4iei aa4aui Ba Kon{Horo B yMoB A-A cxJraAaerbc.ff piBHsHHf, ocnoBoro ,snoro e pinnicrr nacriAox s aftcioMr.r III,: AB = AM + MB. A)

60 = 160; 2x -- 22O;

l.11.OA=OB

Bi4cranefi uinc cycigniuu roqxaMrr Ha o,q[ry Meurue, rlilr{ caMr.rx rorror. 1) = 60 BiOnoaiOa: E) 60 cu.

e l; P e l; S e P8; W e Pqi B e l; <rouxa S.nexrarr nainr toqrcama P i 8>. 6) R e m; C e mi B e MR; M e -41?; <rouxa X se.nemr,rrr uin< ro.rxanrra/. i M>.

BiOnoai1o: B) 4,5 cu n 7,5 cu.

= 16O; 2x

1.1O. a)V

50' 2 = 100'.

1.6.3a axc. TII,AB = AM + MB. AM = x cm, BM = (r + 3) 12 = x 1- (r + 3); 72 -- 2x + 3; 2x = 9; x = 4,5. AM = 4,5 ctr; MB = 4,5 + 3 = 7,5 (cu).

1.7"3 cu.(21

IAOM = ll0"; IPOA = 110' - 60'= 50'.

1.3.A, A. 1.4.8, r.

= 18.

l) AB = AM + BM; 8 = 3 + 5 2) AB=AM + BM;L9 =7 + 12 3) AM = AB + M B ; 27 = 22 + 5 4) BM = AB + AM ; 2L = I + 72 -

piBeHb

liilt{ifli;il

M JreJrct4rb rvrinr ro.rxami,rA i B. npanu.nrno. To.rxaM relrcrarr rrrix roqrcaMl,IA i B. [paBrJrBrro. To-rxa M,nexrr,rrr rrrix< ro.rxandla A i B, npaBr.rrrbHo. To.rxa A aerxarr uinc ro.tcanrz B i M, npaBrrJrbno. To.rrca -d. .nencrzrr rvrinr ro.rxarvrra B i M. rpaBr,rrrbHo. Toqxa B .lrelxr,rrr rvrix to.rxairrr,r-4 i M.

flpaBr.rJrbno. To.rxa

5\ BM = AM + AB; 37 = 2l + 166) AM = AB + MBi 33 = 18 +'15 -

x = 22.

reouerptg sinsxina o. 9. ra ix. Axa,qeuirHri

pirnocrefi:

iiiiiiiil;{il*i IEOMETPI9 Einqxina O. 9. ra iH. Ara,4eMi,{nraE piaenu

!{ti

625

1.13. AO = 18 cM' OB = 14 ctn. = 7 *r. do = 9 cu; ona ' lon ro ,-lZ--2

= KO + OM = 9 + 7 = Bilnoeidu: 16 cur,

x--T---i'

1.14. AO : OB = 3 : 5; AB = 48 cu. 3a anciomoro eumiproeanna ni4piaxia AB = AO + OB' IIosHaquMo xoe$iqient nponopqiituocri l' To4iAO = 3x; OB = 5x; 3r + 5x = 48; 8x = 48; x = 6' AO -- 3 '6 = 48 (cnt); OB = 5 '6 = 30 (cu)' Bi}noai1u: 18 cu i 30 cm.

1.15. OCrti.xrx[ HeMO ABa

B uoXe

TorIRa

sHaxOAI'ITI4c,fl

no o6ugna 6onz ni4 rovxu C' To poBUItl

Br{na{fiu.

(cu); MC = MB + BC (axc.

L) Mc=f,ec=rz,z=o

MB=MC-BC=6-5= I (cu). 2l MC = |eC = e cu. To'{ra C .nexzrs uim MB=MC+CB=6+5=11 (cu)' 1

Bi}noai}u: 1 cu a6o 11

6. Hexafi IMOC OM npoxo4urr,

ro'rnarvrr

III')'

,"+-;tlt:

Bi0noeiOt: E.

llpz neperuni.qBox trp.rMilx a i nuyrpiuHi oAnocloponHi

= 2r' Ocxi'rr'xra npomiur' ctoponarrrz nyra KOC; ro 3a axcioryIoro

1.17. MK = x cv; AM = (r + 5) crl.; KB = 2(x + 5) cxt' AB = AM + MK + KB (anc. III').

xt(xi5)+2(x+5)-75'

x + x + 5 + 2x + lO = 75 4x + !5 = 75; 4x =

6O;

A x = 15'

MK = 15 cv AM = 15 + 5 = 20 (cm)' KB=2 2O= 4O (cn);-4J( =AM + MK =2O + 15 = 35 MB = MK + KB =15 + 40 = 55 (cu)' 2O cnn;

(cm);

40 cl.,r; 35 cn; 55 crt'

1.22. ZB = Z1; l5 = 17, srt< neprrzxa.nrni. To4i, ocni.nrxz Zl = /7, ro ZB = 15, ane qe eHyrpiurni pianocroponni nyrr npu trpsMlrx a i b i ciqsiii c. 3a ossaxorc naparearHocri upeuux o ll b. Bidnoei}u: a I b. 1.23. ZCAB = /2 + ll = 72" + 72" = 144". Ocniasxu Ba yMoBoro CD ll AB, ro Z3 = ICAB aH nuyrpiuHi piaHocroponHi, roai l3 = L44". Bi}noei1u:144".

1.18. Hexafi IAOB - 4aHzii rcYr' IlpouiHr OC npoxo4zrl uint fioro cropoHaMr:

Sosnilreifi rcyr rpfinyrHuxa gopiruroe cyui nuyrpiuuix xyrie, ue cyui*cuux

lAOB=/AOC+ICOB' /KOC

=I2 Zloc,

IKOM

= tKOC +

=f,t^o" 1

3 HUM.

zcotw --L zcoB: 2

ICOM

tcoB)

ztoc

= qt

+L zcon

! ztto". 2

9, Hexaft npaui o, b i c neperunarorl'c;r B r' O' PoarlsneMo npaui b, c i d'.Bo:r.:rl 3a yMoBoo rar(oxc [eperrluarorbqfl B oAnifi roqqi. Ilpunycrzrvro, ilIo q.tr rotrna ni4uinna ni4 rounu O'

npsui b i c marorr gei cninsHi Toqr{r4' rrlo cynepeqr4Tb rrac:ri4ry a axcioura Ha.{e}r{Hocri. Otxe, npaui b, c i d raxotx treperr4Harorbcfl e rouqi O, ro6ro sci 4 npami fleperr4narort'cfl ToAi

e oAHifi

626

b ci.rnoro c vrBoprororlc.E napz nyrin: 3 r 6; 4 i 5; 3 i 5; 4 i 6;

uuy:rpiurni pianocroponni nignonigni 2 i 6; 1 i 5;3i7;4i8; rrosHimsi picaocropouHi 2 i 8; I i 7. ,flxrqo npu [eperr4Hi ABox -trpfiMfix a i b ciqsoro c eiryrpiuui (sonuinui) pieHocropoHni Hyrrz pieni, a6o cyua nHyrpiuuix o4nocroponnix xyr:in cranonnrr 180", a6o eignonigni Hyrz pisui, ro npaui a i b raparemHi (i naenaxu).

cnn.

x, roai IKOM

rvrixc

Bilnoei7a: 15 cw;

t.2'l .72+108'=180'.

rurvriprosanur xytin lKOC = IKOM + IMOC' Maeuo pinHanHfi:, + 2x = I53; 3x = L53; x = 5L' IMOC = 5L"; IKOM = 2' 5t' = 7O2"' Bi}noai)t:51'; 102".

r.--.....

llpz reperrHi gBox lpsMux a i b yreoprororrcx napu cyuixi 2, 2 i 3, 3 i 4, I i 4) ra neprzrcarbHux (7 i 3, 2 i 4) xyrin. iix s,'racrzsocri: L Cyma cymixuzx Hyrin gopinuroe 180". 2. Beptr.rxarrri xyrr piaui. rrux (1

16 (civt)'

1.2' Onopxi Sarrr Kypcy nnaHiMerpia

ro'{qi.

piBeHb :{ii;$ii:iil {:it rEoMETptg SincHixa 0. 9. ra ix. AxaAervriqHrr

1.24. IKMN = x = lMKZ + ZKZM = 72" + 63' = 135'. Bidnoaidu: 135". 1

.25. floanaqnuo xoe@iqieirr nponopqiitirocri r, Togi ZB = 4x; lA- ZC - 3x (Hyrrz npra ocHoni pirno6e4penoro rptr xyrnuHa pinni).

3x + 3* = 180; 10r = 180;

zB ="4 .ta" =

72". ZABD

= 18.

=1ru 22 =I.72"

=36".

Bi1noeilu: E) 36". 1.26. B;

I; E.

i:ii:iil:ili;,:t FEOMETPIC Einsxina O. 9. ra in. Axa4ervri.rHvri piBeHb

iL::l

627

1,27. ZAOB = 180'src pogropgyrrrft; ,/AOC + ZCOB= 180'src cynixni; ICOB=18O" - /AOC = 180" piBno6eApeHufi (OC OB nK pa,qiycn). ICOB - = (180" tCoB\ :2 == (180" - 132') :2 = 24". tC = ZB -

1.36. r, = Q,$1. r * 0,5r = 18; 1,5r = \8; r = 12; 0,5'L2 = 6 (cu). BiOnoeidu: B) 1-2 cu i 6 cm.

4a" =132".

Bilnoai1o: B\ 24"; 132; 24".

nf = 9n; r = 3 (r > 0). l=2nr=2n.3=6n(cn).

1.37. S = nPi

Sa.nenrnicrr rrirx pagiycauu -R i )fi aerbc,fi TarctrMr?I tpopuy.nauz:

r ta

cropoHoro

rl [paBnJrbHoro n-rcyrHr,rna

Br,rpa-

Bi0noeida:,4) 6n cu.

1.38.a,=2ri

zrJB

RJ'

,Ji

RJ'

2rJg

2rnl3

ftJB

r=?,2' ,=9=t." =nf =n.3,=9n(cu,). 2

Bi1noai1u: A) 9n cu2. reopeMoro flitparopa g aBDC: BCz = BDz )_ DC2; BDz = BCz * DCz.

L39, 3a

BD = \@ 4 = J6-4X5-4) = Je = B (cu). AC=AD+DC=2+4=6(cu). s^,""=lec.BD=+.6.8=9 (crvrr).

1.28.

R=o=!6

'1.29. P = (6

Bidnoaila:

1,3O. Biinoai1a.'

+ 8) . 2 = 42 (ctt). 42

1.4O,

crvr.

A) 1 i

Bi|noei|u: B) 9 cn2.

I I auna0orc. Ana-norivno,

tOKC = a ODC sa rinorei rocrpaM nyroru (OC cni.nrxa

euna1orc.

uyeorc

rinorenyaa, OC

DC = KC = 6 cu; AC = 2DC = 12 cu.

AB=BC=11cu.

6icercrpzca Kyra C).

"Ioai DC = KC = 5 cm; AC = 2DC = 10 crq. AB=BC=6+5=11 (cu). P,*"= 2 11 + 10 = 32 (clt).

. 11 + 1"2 = 34 (cu). Bi0noai1a: B'1 2 a6o 4.

P^*"2

5cu 1

.31 , d AOB

rp.EMoxyrrrnirt AO L AB sx pagiyc, npoBe4errlrfi y rorrry 4otraxy. Togi

Oe = 1OB= 2

6 cu grc Rarer, Iqo rreficr,rrb

TrporI,I

Kyra 30" d = 2r = 2OA= 12 clt.

Biilnoeila: E) 12 cu.

1.82. a=2ort2

co";

Bidnoeilo: B) 10rE Cepe4ra 'l

2r; r

=t='!P=

10JZ (cu). 1,41. AC = 2R g'r<,qiaMerp RoJra, IABC = 90o src nnrcairlrft xyr, r{o crurpaerbc.s ira 4iauerp.

crvr.

ninis rpanerlii gopinnroe nincyui ii

,33. Ilosga.ruMo AoBlnlrgy uenruoi ocnoBrr PeAH.SJIIHIS:

,r+(.r+8)= 2r+8 ,

Ba .r cM, NlaeMo:

=I+4.

BC = AC . sin

ocHoB.

(r + 8) clt. ToAi xL4=10;.r=6. 6i.lrrtroi

Bidnoeida:

LA = 2rR sin 60' =

oo lo

"i"

= nJa.

-Rv6.

ce"

Bi1noei1a: B) 6 cu.

=!0,o,, 2'" s=12 ro.86=r8o Bidnoaila: B) 180 cm2. 1.35. Y dABC: /A = 25" (ei,qnosi,4ni),

1.g4. s

lC = 4O'

(cru,). n

Saaerxnicrr ir.rirx croponoro d i,qiaroHa,rrro d rnagpara:

(aepruxa;rriri),

tB = L8O" - (25' + 40") = 115'. Bi4noeidu: A) 115'.

628

fiti: TEOMETPIS rincxiHa O.

1.43. a = 2r = 2 . 5= 10 (cu). it = Bi|noaidu: f) 10rE cru.

ra in. Ara.qewrivHhr piBeHb

ilitltiiiil:

il:iliiiil:li

L}r[i

FEOMETPI9 Eincuixa O.

a=a"li

cv.

f . ra in. Ara4ervrirxraii piaexs :i:iii; 629

,{ra rpuryrnuru no4i6ui,

1.5O. Ilponegeuo ni4piaox AC. arcrqo:

rcyrir a,AJ{C 180", ro4i ZKAC + IKCA= 180' - 96" = 84". ZKAB t ZKCD = L22" + 142" = 264". ZBAC + ZDCA= 2640 - 84a = r8Do. A"ne qi rcyru niryrpimHi o,4uocropoHxi npv npauux -AB i CD i ciqsifi AC. Ortte, AB ll CO. Bi}noaida: AB ll CO. Cyma

1) gea xyrra oAHoro rpr.rxyrunrca ni4uori4uo piBHi ABoM nyrarvr iHlroro; 2) 4ni cropoHu oAuoro rpuxyrxura nponopqifiHi 4aou cropouaM Apyroro' a Kyrrr' yreopeai urrMil cropoHaMra. Pieui; 3) niAnosi,4Hi croporrr4 rpzryrnizxis nponopqiiini. flpama, flKa neperr,ruae 4ni croponrz rpl{rcyrauna i napaxe.nsua rpetiii croponi,

ei4runae ni4 nroro rpr'IKyrnr{rt, troAi6Hufi AaHoMy.

1.44. Ocximnu AB I DM 3a lMoBoro; ro IBAC = IADM aH rgyrpiurui pisgocroponni (civna AD). IACB = IDCM flK BeprrrarlbHi. Toai yABC * aDMC 3a ABoMll

1.51. Hexafi AM = x cm. ro4i BM =(x - 3) cu aa yMoBorc. BK = BM, AM = AZ, CK = CZ ar ei4pianz Aornqrrux, npoBeAeH[x s o4niei rouxu Ao riorra. Tottty AZ = r cM, roAi CZ = (x 4) cu. P"*"=AB t BC +AC=2x-2(x - 31+2(x 4\=2x+2x

IIYTAMII.

g, Ac-=1, Ac-3'1'5 -4,b (cM). | cD- MD 1,5 1

3a yuoeoro 6x -.7O = 28, asiAnr

= AC + CD = 4,5 + 1,5 = 6 (cru). Bidnoei}o: B) 6 cm.

AM = Az=

18.

,(iaroHa.ni pou6a nepnen4ury.rapui. ,(iaroHa;ri pom6a neperr'firarorbcs

l$*liiii:

iiili::::::::i:

roqKoro

BC2 = BO'z

OC2:

[iii:$ji:i

(irvr),

I 1

IlosHaqnMo MeHrrry cropony napaJreJrorpaMa

cM' a cymixcny 3 Hero

(x + 8)

civl'

Ilporrznexcni cropourr napaJreJrorpaMa pinni, romy p = (x + (r + 8))' 2 = 4x + 16' Ba yuoeoro 4x + 16= 48, aeiAxu 4x = 48 16 4x = 32; x = 32 i 4i x =8' Mesua cropoHa flapaJreJrorpaMa

Bidnoai|a: A) 8

1.48.

cn.

aABC-pieHo6e4peHufi,

8 cu. B

AB=BC,1BCA=IBAC

slr Ryrrr npz ocroni. IMBC = IKBA, 6o xorxnrzii is Hrx e cyuoro ryra ABC i o,4noro s pinunx ryr:in

ABM|CBK. aMBC = dKBAsa cropoHoro i 4oova nprrrerraMu Kyrauu. './ 1.49. IABC - npnMoxyrull frr (AC LCB sa yrvroeoro) 3a reopeuon fli$aropa AB2 ' AC2 - BC2;

n / / ,/ / ,/ / 5A

Au=tpiz"=,lisnuq=t5oe=13(crvr).

Bi7noeida: 13

630

cvt.

6:.

t"*t. x-4

tcnr).

fi=sJ5 clu; a=ar6=sJ5.16=g d s 9./s 3./3 f=2,lS 2J3 23 2 tG Bionoeioa:

Bc = Js' *r2' = Jsr.M4 = Jr% = 15 :rj:+::x::::!:: P^.-- , L5. 4 - 60 (cM). '.{i:i::.:+..ri d-iioaiao: E) 6o mr. 1,47.

10.

(crvr). AC =

u*-r

Zx-4

ri = sl

rc*1.

1.52. [un..ra6.n. ao Nb 1.28.

18 cv, AC = 24 cw, ro4i Bo -18: 2=e(cu); oc = 24 : 2=72(cm)3 o BoC (to = 9o') Ba reopeMoro fli(paropa:

S;1:;11:;;;: 1.46. Hexafi BD

:l:i:li:i:i:fii

4i,rarrca nannil.

6'2x-8=6x

(cnr. BC=s!*21 AB 6r-+B]=92 33333 Bidnoeida:9-2 s? .*, 8? 333 "*, "*.

2On=36O;n-18.

neperr.ruy

Croponr.r rpuRyrHHKa:

1.45. 180(z - o) : 160n; 180n - 360 = 160n; 1802 - l60n = 360;

6r - 38; -

o] t"*t. BM = BK = u}-

KC=cz=u*-n=zj-

Bi0noai}t:

\ \ \ \ \ \ 4 C \O lffiq. / \

l: \

\ l.:) \ l-/

iiii: TEOMETPIS Eins*ixa O. 9. ra ix. Axa4ervrivHrf piBeHb :ii:i:i:i:ii::#:

a=2r.1!;

"*.

MolrcHa Bnr{carr4

Koro, sxqo cyma fioro rporrz.nemurx cropin

prBHl.

1,53. Hexafi BE = 7 cM pieuo6eApenoi rpa- Bficora ne4ii ABCD. 3 aABE (lE = 9O") AB = 2BE = 74 cut. (ICarer, rqo JreJ(rrrr, rrporn ftyra 30o, gopinnroe no.noenui rinorenvau.) Cepe4Ha .niHis rpaneqir 4opinuroe nincymi ocHon:

BC+AD 2

A-ne

ocni,rrrr

n rpaneqiro Mo'fiHa Brrrrcarr4 Ho:ro, ro BC +

= 2AB = 2 . 14 = 28 (cu). Orrxe, cepeguR

linia

rpaneqii 4opienroe

Bi}noaiOu: 14 cm.

1.54,ABCD

\-Y

V vorupunvrHrzrc

@wt);

AaHa rpaneqia,

AB = CD; IABC =

AD - AB + CD = 2A

i =ta

(clc)'

L5O".

Cepe4Ha.ninia 4opirnroe 16JE cnr. BE LAD - Bzcora rpaneqii. AD + BC = 2.16.yE= S2aE (cm). Ane AD + BC = AB + aD (gvs. safla.ry 1.58). A AD !- BC

Sl;iscu '' AB

- 16rF

cv.

ii:iilli;i::::.:i rEOMETPI9 Einsxina O.

ra ix. Axa4enivnnfi pieexu:.{il 631

V aABE

= ta0"

150' = 30'. BE

=;AB

Bilnoai}a:8Jg

-- 15 cM

aABD: eO = ! nC = 8 crvl; 2

AB2 = AD'z

BD2;

3 rACM

AB ' sin 45"

Ba reopeMoro

= |O\E 4 J2

CD=5-20-25(cu). Bi?noai1a: 20 crvr i 25 cv,

,flxrqo e ro.rrrcra rroaa KoJroM rrpoBeAeno civuy i 4orra.rny, To lcBa{par AoB}nr,tnl4 Aornvnoi 4opinnrce Ao6yrrcy ci.rnoi sa ii sosHiurHro qacrnHy.

Bi1noai1r': 24 cv,. 1.57. ABCD

poIvr6;

AD = lO

cwr,

AOz

oD = $o,

.t oDzt oD = ^[ADr-

fioo -

s^ ^^^ = L

22ao. oD =!. A.ne S*o, =Lln'nrt

BiOnoaida: lO cm.

AC = 12 cla. qe Br4cora aAOD

3 aAOD(IAOD=sO',AC IBD\:

OD=LeC=a

6. 8 = 24

OK.

1.62. Hexaft MC = x cu, ro4i 3114 = (15 - x) MC2 = MA . MB; x2 = (x - Z). (lb - r); A

(cNr).

(ew2).

oK=r--#=T=4,8

(cu).

Bi1noetla: 4,8 cut.

1.58. Ilpone,qevo OC L AB, togi OC - ri4cranr siA I{eHrpirno6e4pernfi. pa Korra O 4o xopguAB. aAOB

- Bvcora OC paAiyot xo.na. AO = BO = 15 crvr e rvreAianoro: AC -- BC. B rAOC: AO2 = AC2 1- OC2; Ac = ^[Ao' =

ftb

-oc' =.Eb' -tr'

-1acb +1r,

Bilnoai1a: t8

&.n

crtt.

= AB. AD.sir,./A= 6.16.sin60'

96 f 2

.63. 3a s,rac'runicrrc Aorr,rrrHr{x, npoBeAeHr,rx a odniei roqrr xotra: AD AE 6 cut; ' BD BF g ctn;AoCE CF x= cttr. = = = = = AB=6+ 9=15(cru); AC=(x +6)cru;

s-,. 6

48JB

AC . BC = |22

Bi1noeidv: 54

60'

cru E

cnr

(cu'?)'

a co

.g .tz = b4 (cur).

cwr2.

(BC ll AD), BC =8 cw. AB =6JI 1,64.ABCD MeHrtra cropoua - rpaneqir "r =- 12O", Ilpone4euo ,sr cni.nrnuft fiepnerrAnrdyJrsp ABox napaJreJrbHnx fiprrMt{x . lC CK L AD, rogi CK = AB = 6& cw, AK = BC = 8 cM sK rporr4J,ren*ri cropoura npamoxyrlauna ABCK.

Bi\noei\a: 48JB cvl'.

692 l#j reouerplf

=(r-2)cu.

BC=(x+9)cm.

AB2 = AC2 t BC2; 152 = (r + 6), f (r + 9)r; E 225 = x2 + l2tc + 36 + x2 + 18r * 81; 2x2 + 30x* 108 = Oi x2 + !5r * 54 = O; Jcr = 3; tcz = -18 yMoBy C - rre Baa4oBoJrbr{s€ aa,qa.ri (r > 0). CE = CF = 3 cnt. AC = 3 + 6 = 9 (cu); BC = 3 + I = 12 (cru).

= e (cu).

parruir rapaJrerrorpaM. ZA = 6O" ' 1.5g. ABCD 6icexrPuca lB. AE = 6 cm, BE ED = lO crq, AD = 6 + 10 = 16 (cu). IAEB = ICBE ai nnyrpiruni pissocroposgi npuAD ll BC i ci'rnift BE. To4i A IABE pisHo6eApelruii, AB = AE = 6 cut.

ctu,

. x2=l\x-x2-30*2r; ' xz I x2 - l5r + 3O - 2x = Oi 2x2 - 17r + 30 = 0; D = b2 - 4ac:= 172 - 4. 2. 30 = 289 - 24O = 49; L7 !7 xr.r=--{-t xr= 77 -7- = 2.52 x^" =17 *7 = 6. 4 4 7) MC = 6 cut; BM = 15 - 6 = 9 (cna); 2) MC = 2,5 ctn; BM = 15 - 2,5 = 12,5 (cm). Bilnoeila:6 cnn i 9 cu a6o-2,5 cn i 12,5 cn.

"*;

Ao,;

86 =.uGZ =

x"--(x-5).(x-5+x+5); x2 = (x - 5)' 2x; x2 = 2xz - lOxi x2-l0x=O;x(x-10)=0. x = O He Ba,qoBoJrEI{ge yMoBy aa,qavi (r > 0); .r - 10 = O; r = 10.

4'6 = 24(ctl); "86'36 =

Pagiyc nnr,rcauoro n pou6 KoJra

ADz

1.61. MC = r c]Mj AM = (r _ b) cM; AB = (x + 5) cna. MC2 = MA. MB.

= r0 (cvI).

fli(paropa: AC2 = AM2 + CM2;

cM =,[Ac, - AM" i cM = Jze -to'= !tr6 -10Xr6 +10)

cru.

AS=BS=10crulAB=2Oc.lq

1.56. Ilprrvrirra. V uigpytnury Aorryqeno noMrrrrrcy' ts'yuori rtae 6Yrra AC..= 26 cu. aABC, AM L BC, AB = 101E cw, lB = 45'. =

- aagairi xop4ra. cepe4ura xop4u AB, CS = 5 cu, DS = 20

AS.BS=CS DC;r-x=5.2O; x2=lOO;x=1O(x>0). '.

ocuoBa.

*tb" = J64+2% = tE89 = 17 (cu). Au = 'le' Bi?noeidd: 17 cw,

3 aABM: AM

neperlrnarorbcfi e rouqi S, ro AS . BS = CS . DS.

Hexaft,4S=BS=xcu.

= BC'1. Blrcora i Inegiana rABC. AC = 16 cu

/3 i CD ro;ra

1.60.,43 i CD

cM.

piBuo6eApeHuit (AB

1.55. IABC BD

,fkcrqo xopgn

8Jd cu (ar xarer, uto rrernurb

nporr,r rcyra 30o).

sincHiHa O. 9. ra

ir. AraAeniqHri

piBeHb

il{til{lill.l

il}:ii:iii:i:l FEOMETPI9 EincHiHa O. 9. ra

iH- Aragervri.rxr,rfi

pieexu

ii:i:

033

IKCD = IBCD - 90" = 120" - 90" = 30'. 3 :CKD: KD = cK ' tg 30"; KD = 6J3'* = O t"*1. AD

s.^-^ =

+,14 BC' M .cK 6'6 =8

G6'6

B aAoK

(cm').

BC I AD;

BC = 14 cttr, AD = 4O crr.. CE LAD, \i Br{cora rpaneqii, AC CE = 39 cM - 4iaroua.nr. Ocri:rr,nu EaBKoJro rpaneqii-4'BCD orrucano r{oJro' qe piaHo6ivna rpaneqia. ognaqae, uo ABCD 14 (cM);

Toni -22ED-AD-BC

-40

^.CED:CDz=CE'+

s*,, A

s..-^ = f a-n.cE. 1.40.89 = 280 (cv,). 22 ocni.ntrz on[caHe

HaBr{oJro

rpaneqii

KoJro oAnoqacHo e onrrcaHuM HaBKOIO

(cu). (cm).

aACD'

AC 92'AD ro fioro paAiyc li no:nna gnaiittr a gopuy;rn: R ' 4S

^ = 15Jto

.13,/10 40

4 .780

-25

(BC

ABCD

3 IACF: AF =

AC2 = CF2

AD). AC

15 cm; BD

Bi0nosi0a:150

=ry.t2 2

,zoAD =

= Ao.ts30-=

tl1-

-d,d, = ;o"

= + = i, ?! oA722

t, f

- ar6 tmt), BD = 2.oD = sJ5

= "o ;.24.8J3

96J3

(cu).

(crra').

cm'z.

77+25+28 n= AB+BC+AC=-z =35' S*" = J p@' - o)(p - b)(p - c) = JBb L0'?. 18 = 210 (cM); Are S*" = S*o r Sn"n = AB. Ko - + BC .oF, 22 A ae KO = FO = R - pa4iyc ganoro xona. Macmo: ln zs*\n.tz =zto, 22 l=

3a reopemoro xocunycin: 2BO

2O cttt.

AB'= BO' + AO'

= ^ln{-r44 = Jal = g

l5o

Toai AB2

(cm).

-4-8'?

BC2

- BC' =

4AO .

-4. AO.

BO.

cos

oY 60"

60o;

2AO BO

6cu

-\,

42;

2AO . BO = 36

16; 2AO

= 20.

f,*= r0r,E (cu,). = ro "o +

91,3. 3a.qa.ri i ueroguii po3B'e3yBaHHe 1,7O. Bi}noai1a.'

$?6 -5?6= Ji00

60'.

Bidnoaida: 10.,6 cm'.

6) Ana.noriuno. AC = 30 cru, BD = 26 cv, BE = CF = 24 cvr. B d BED: ED =,tED' - BE" =.1T6' 24' -

cos

s*," =L.2Ao.2Bosin6o" =2Ao

(cm,).

cu2.

.AO.

IBOC=180"-60'=120". BC2 = BO2 + Oi, - 2OB OQ. cos 120. BO2 t AO2 t 2BO . Ao.tos 60".

Bpaxoryrovz, qo BC = EF, maemo: AF + ED =AE + EF + FD + EF -AE + EF + FD + BC =AD + BC' Ssitcu: AD + BC = I + 16 = 25 (cu). s-- - --rpan.

sin

1.69. ABCD'- trapar'rerrorpaM, y snoro AB = 4 cm, BC = 6 cttr. flporu rocrporo xyra uirN 4iarona,raMrr relr![Tb MeHrrra cropoHa: ZBOA = 60",

l AFz;

$e -cr" =,llb" -rr'

^AOD,

szipnu R(12,5 + 8,5) = 210; R = 10 (cm). = nR2 = n. IO2 = 100n (cu,). Bi0noeidu:100n cm2.

(cy).

= = - rpaneqiaBr.rcorr4llrpaneqii; BE = CF = 12 cttt. CE L AD, CF L AD 3 aBED: BD2 = BEz I EDz; ED = JED' - BE' = Jf -tr" =.,@o -M4 = Ji56 = 16 (cu). a)

Bucola

S..-_.__

Bi1nosi}a: 25 cv.

1.66.

l.68.3uait,qeMo nrorqy IABC ea (popuy.rroro fepoHa:

n' =,ttsz:]+ zzs = JrrSo = rr&0 ED2; CD =J3y -1# =rGr1-16' =JG90 =13f0 = JSr,

= 9o"\:

Bi|noei|a: 96rE

=t3

13 = 27 (cu). 3 IACE: AC2 -- AE2 + CE2; AC

AE = 4O

1,zK

3 .AOD: oD

sa"qarra rpaneqis+,

pon6. OK L AD

ro6ro ZOAD = 3O'.

Bi|noei\t: 6616 cn'. 1,65. Hexafi ABCD

oK = 6 cv = r. AC = 24 cv:'2Ao =l oc =r, .*.

AI< + KD = 8 + 6 = 14 (mt).

1.67. ABCD

10 (cu).

f) {r + 5\ . 2 = 2A.

1.71.

rACF: AF =.[Ae 4F" - $o' - 14'= ./600 - 5?6 = JEr4 = 18 (cu]. AD + BC = AI + ED= 18 + 10 = 28 (c*t). B

s^-^-rp^n -

2,8 2

.24

336 (cn'z).

1,72. Bi1noeida:

Bidnoeido:336 cu2.

634

i::::::

FEOMETPIS EincxiHa O. 9. ra in. Axag,ervrivHrf piBeHb

:iiii:iiiitiii

i*liiiitiiii

TEOMETPIC Sinsnixa O. 9. ra ix. Axa4ervrirur,ri pieexr ll:j:

635

La2. ABCD

1.73. /AOC = )ci x + 17r = 360; 18r = 360; * = 20' uoei,4uoro qeHrparrbuoro xvra AoC,

to IABC =)z'+oc =f;

'o"

= 10''

1,74.ltarer , npra.nernraft,4o r{yra e 60" e [porr4Jre)rcHuM Ao Hyra 30o .nosrzui riuoreuyer: 10 : 2 = 5 (cu).

=!dC 2"*

cw.

s*"o

P"*,

= AB

1.83,

4 AH + BHi

BH3 AC8

Bilnoai}a: B) 45

= 45J,

'l,t2

crvr'?.

;AC=4

cmi

oo =.f,ao =z

cM.

....-

6:2 = 3 (cm).

.,6t-

HaBKoJro rrpfiMorcyrnoro

N (cm).

rpr,rxyrnar{a, Jreficr,rrr, tra cepe-

O 4anoro npsMor{yruoro rpr.rryruraqe qeHTp onrrcauoro KoJra, rouy OA = OB = R = 17 cru.

AB=17.2=34(clr).

BCz;

to, =

=*.16.80 =

"fu4

g0XB4

- s0)

,14.G4=2.8=16

(crrn).

240 (cm2).

,{,'ra poar'asanna qiei sagaui cftopucraeMocs MeroAoM nJrorq: nJronly rpr4ryrrr[xa anaft4ervro aa ipopivry.roro lepoxa, a noriu, osaroqu nnouqy i cropony, s (bopMyra 1

'2 =ah

anaftAeuo AoBlrcr,rHy sncoru h.

1.85. flano: IABC; AB = 29 cu, BC = 25 crvr, AC = 6 cm. BD - nucora, [poBe,qeHa 4o menuroi croporlr AC. Ba Sopuy;roro lepona:

(cM).

AB+ BC+ AC 29+25+6 60

=t=30

(cu).

=&s.L %=J6 b.b'o4 =6.5.2=60(crur). s*" =!rc.8D; BD = = zo t.*1. #;= #,

Bi0noei1u: 2O cw.

nB. AC 6 10 = on = + = tb (.*).

feOUefRlg SincHiHa O. 9. ra in' AraAeilri'{Heril

#=f

Bicercrpnca Kyra rp[Kyrur]rrca qifi ni npu,'rer;rraM cropoHaM;

BiAnoBiAb: 15 c:[{.

i:1:

=,lrs-s =.,,[o = a 1cm;. =,t@ + BC' =.[1" - g = fi6+ g6 = Jaz = zrTs

g^ =

Bi}noai}a:. B) 2,4 cu.

636

cwr.

=|AC

s*" =!nc

AD='[4\g" =Jro-9 ='l%=5 (cu). 11 oD = n' t = 6 (cm'z). s*oo = ;' iAo. 2s*o, _r_.p r -rAD 2,4 OKi OK= S*oo = b b =y= AD

a rDBE, r"*u

= 9O").

ue4iana, AD = 5

Bidnoai1o: 24O ott2.

pagiyc nnzcanoro r(orla. OK L AD, OK -- r - AD2 = AO2 * OD2; 3 IAOD (ZO = 9O'):

ssinxu AB

Ac =,IAts" - Bc, = $nz

-1

aABC

(lC

up,nMoxyrrrr{ft.

cepe4rana.CB, roruy CO

64JE (cu').

cu'z.

3 yABC: AB2 = AC2 *

lcwr'\-

1.8O,HexaftAC= 8 cn; BD= 6 cu. ToAi AO =

1.81.

=L#= + Js

rc*1.

1.84, Cepeguna rinorenyera

1.79. Bi\noeida: l).

BC = 8'

I]errp roaa, onucalroro 4uni rinoreuyar.

(cm).

. sin 45"

8cu

aABC: AB' Bidnoaila: 2Ji3 cu.

a 24 24Jg R= F= F = ._=8J5 Js J3 lJ Bidnoeila: A) 8G cu.

li{$:r.:*.:

^ABC

1.77.

$11tr1:i.J:

3 aACDI AC=]AD, -CD ''z =

1.76. Bilnoai}a; B) 2, 5 i 6.

s.;Jr.

BC = 6 cm. AD

.78. s =

Bi|noai|u:64Jd

Bt}noailo:1.

BC=*-!+*=-+= =!Z=?=4 cosZCBF cos3Oo .

cm.

BH; BH = 12* 9 = 3 (cm).

72 = 5 + 4 +

BaBCF:

4opirnroe no"

pisno6eApegzft. Ba ynonoro AB = BC = 5x; 1.75. rABC AC = 8x; 5x * 5x + 8* = 18; l8x -- 1'8; x = I. AB = BC= 5. 1 = 5 (6r); AC = 8' 1 = 8 (clt).

,AA

ryrin

3a axciouoro nrauiprcnannx

ICBE - ZCBF + IFBE; ZCBF = 90' - 60' = 30".

Bidnoaida: B) 70'.

Bilnoaidv: B) 5

Aanuft napaJreJrorpaM. flpone4ervro

BE L.AD, -BF I CD * Br{corra raparreJrorpaMa. BE = 8 crvr; BF = 12 cv.. ZEBF = 6O. ICBE = 90'.

ZAOC =2O". Ocxilrxr.r rpagycna rripa nllucaHoro xyraABC Aopinnroe no.nonr'rui nig-

piBeHb :i:iii:i:i:i::ii:

4irurr

rrporr{Jrextuy cropoHy Ha

rigpienn, trpotrop-

l:*l{iiiliitrji TEOMETPI9 Eincxixa O. 9. ra ix. Axasenirxraft pieenu {{";il 69z

1.89. 3a Hac.nl4nou g trpaBuJra

lC = 9O". xyra A. CD = 24 cu, BD = 51 cu.

rpr{rcyrHr'Ix' .86. Hexafi -4BC - Aauufr -AJ)

= :UB '2'22'

-.6icexrpuca AC 24 - :. Hexafi x - Koe$iuieHr AB = 51

lVlaevo:'::

+ 752i

, Fro*, =f,.

26OLx2

576x2

?=40

s*"

Bidnoaidu: 1500 cv2.

$$liiliiii

IADB).

6O cm.

atb

trpaBI4JrO TpI4ItyTHr4I{a:

,qn+nC=dC:

638

*:::: FEOMETPIS EinqxiHa

M: *

CA r BC r

CB) =

cB)=' .0=o. * =''

;x',

atJ-lz

BeprrrzHr Maru

o*o o-b . b, u(o tb). \2 2l 2 =9, 2 ",= 2 2 t

cta

ToqxaN: x-al

=2o ",,=b-b =b, N(?!*",a]. 22"2\21 Toqxap: r_a+c+2a+c _3a+2c. u=b*O =L. ,( Ao*2".!-). 22"22\22) To.rxaK:

., -o+2a+ c ='o! 22\2)

o; x(2o! ",o\.

,{oregemo, rqo MNPK - traparerorpaM. KqopAzsaru cepeynHv giarouani MP:

o," = 74 + 22 + 24 = 60 (clt)' ^

x='

a 3a+2c ,l c

2a,c

b ,t, '

i u

22"22

b h

ICoopgznaru cepe,qrrurl,4iarona.ni NK:

^*"

2a+c+ 2a+c

2a+c: u- b+O 2"22

A(0;0)

Koop4rznanr a6irarorrca, orrxe, 4iaroHali MP i NK treperuHaTorbcfi i tovrcoro neperrany 4i.namcs naeni.n, A 4e oanavae, ryo MNPK naparerorpaM. flopinxanuu a6c4ucu roqox N i I{ (eoxz pieni), po6uuo BrrcuoBon, u1o 4iaroua.nr

NI{ raparemua oci op4auar. Aua.norivno, opAunarrz ro,{orc M i P .qopieHrororr 1, 2 rony MP OX. Bucuoeon:4.iaroHari napaJerorpaMa trepfleH4hTcyJrspni, a qe oauaqae, qo MNPK pon6, geox cycignix cropin ea Morxna raxo;x o6quc,rzrtr AoBr+(ilHu

/\ \b

v\ aGc

AC. BA + BC - CA t

tDopmyrn KoopAilHar cepe.[ilHr4 eiApisxa:

Cyua eenropin: B

-,/

Toqxa

1.88. Asanoriqno Ao 1.87. 3 aABD 3a reopeMoro (ocrHycrB: AB2 = AD2 + BDz - 2AD . BD cos o.. 3 IBDC: BC2 = BD2 r DC2 2BD ' DC cos(180o c). fligcraniruo BHaqenHfi : 142 = 142 * x2 2 L4 x .cos cx; 222 = 14'? + 12 + 2. 14r ' cos o. A ,{o4arro noqreHHo 4ei ocraHHi pinHocri: 2x2.1'!42 = 222i 2x2 + 196 = 484:. 2x'z= 288; = 14 + 22 + 24 = 60 (cm)' x2 = 144; x = L2 (x > O). AC = 2 72 = 24 (cxt). P

Bi}noaidt

CB).

Hexafi MNPK {oruptrnyrHux, BeputrHx rKoro c cepeailHaM}, cropiu rpanerlii ABCD. Snafigeuo ix xoop4unarlr :a $opmynaMr4 noopArnar cepeAuHu ri4pisxa.

cv. te '' AD=:AC=72 24 cu 2 (1) x2 =722 + 142 - 2'!2.L4 cosa 3 rBDC: BC2 = BD2 + DC2 - 2BD'DC cos(180'- cr). (r + 8)'z = 142 a 122 + 2 72'14 cos a. (2) flo4amo pirHocri.(1) i (2): x2 + (x + 8\2 = 2. (12'z + 14'z); x2 + x" + 76x + 64 = 2' (L44 + 196); 2x2 * l6x + 64=288i 392;2x2 +'l'6r- 616 = 0; x2 + 8x - 308 =,0; D -_ b, - 4ac = 64 + 1232 = 1,296 = 36'; R-36 36 <U-HeSa,zIoBoJIt'HfieyMoBy' -l4i xz---8 *,= 2

AB = 14 cm; BC = 14 + 8 = 22 (cv). Bir)noeida: 6O cv.

BC), CC, -':(CA +

1,9O, Posl,ricruuo.rpane4iro ABCD e cucremi RoopAruar rax, rqo6 Hoop4ruaru: .4(0; 0), B(a; b), C(a + c; b), D(2a + c; 0).

!,*.-.......-.r.'

i@d+

(cu'?)

tpranyruizn, 1.87. Hexafi ABC - lanzfi itoro me4iana, AC = 24 cut, BD = 14 cu flossaqzuo AB = x c1a', roti BC = (r + 8) cm. IloaHaqznro 4as apyuuocri IABD = ct' rcfi IBDC = 180" - cr (ax cynixuzft z 3a reopeuoro rocnnycin s tABD: AB2 = AD2 + BDz - 2AD ' BD cos ct,

rrapaJTeJrorpaMa:

ui pieuocri:

'2' t Aa,

1_t

=i"" "" =; 4o 75 =1500

BB, =

+ BB, * gg, = ::(AB +

=''l-'t.1taB+ BA

5625t 2O25x' = 5625;

(*r).

+ AC\,

,{o4aruo novrTeHHo

nponoprliituocri. Togi AC = 24x cv, AB = 5Lx cv' c Bpaxonyrouu, qo BC = CD + BD = 24 + 5L = 75 (cu)' cr(raAeMo piBHsHHfi. Sa reopemoro fli@aropa,4-82 =AC'z+ BCzi (51x)2 = (24x)2

t_t_t

,[*" - *,1' * 1y; u;'

MN= - npaB[Jro

napaJ]eJorpaMa.

dn+dn-AC. Hac,riaom: ''

AO l(AB 2

2a+c=a\? r. b\'

lto+"\' tbt2 z l*lD-tl =!t z)-lz) ' .

MK= + AD)

O' 9. ra ix. AraAervri'fHnf piBeHb ::iiiiiiiii:ii:ij

SopMyrom:

uIo \.

ql 2)

MN = MK.

Bucnonox: y fiapaJ'reJrorpatta MNPK nci croponu pinHi, ro4i MNPK

iiii;:ii!{till TEOMETPI9 EincxiHa 0. 9. ra ix. Ara4euivrrafi pieexu

poru6.

lif

639

19, /ACB - 90" enncaauir, lqo cnzpaerbc.s na xop4y-48. Tonry.43 4iauerp xona, AB = 2R.

1.91. Poslficr]rMo cr,rcreMy KoopAr{Ear rardvM vnnou, rqo6 fioqarorc KoopAnHar a6irascs 3 qeHTpoM BrrflcaHoro B KBa'zlpar rcoJra' a

3a reopenoro lliQaropa AB2 = ACz L BC2, AB = .lTb\ B" = Jrzb - 64 = .,489 = 1z (cu).

cro-

pour4 rcBa,[para 6ylu napale;rlni ocsu noopAHHar. Bisrueuo sa rconi goeilrry rcqxy A(x; A), orrycrlrMo a nei nepnen4rar(yJr*prr AB, AC, AD i AE rra croposr rcBaApara' ae B(-R; A),

-- 2R2 +

4R2

r. A.

8. E.

14.

11.

xyrir.

""p**.o""r.rx xyrie 180'. cyui*uiax

r<vr'

12,

- 120'=

pirni'

cnn

xop4a.

Ko,'ra Ao xopAr,r.

= OC2

=lE+

4" =

do -

AC2i

16 =

pa4iyc BnrrcaHoro KoJra, a06

lu = 6.6 crnr. z'f 3

a"la;

rco.no,

u{t;, t 2J3

cropoHa nrecrranyrnuxa. 3a

(cn).

roAB + CD = BC + AD.

I - roe(pirlienr nponopqifinocri, ro,qi /ll 7x c;la, CD 9x cw. Hexafi

Cyua

CepeAus

Zl i l2

ainia rpane4ii o6qracnroerr,cs

Ba OopMyroro

BC+AD 3a BCIAD ^ yMoBoro 32. Z 2 = 7**9* 16* Tooi'B*cD =82, =82; =B2t 8x=32;x=4. '222 AB =7 4=28 (cut); CD-I.4 = 36 (cnn).

60"'

II{o6 anafirr,r ai4cornoae ai4notreunn attrcra q' Ao trrrcJla b, tpe6a o no4i'nlrrta na b i oAepxane qficro B].rpasr4rra y rigcotxax (y 4ecar:xonomy .qpo6i KoMy nepe$ecrr4 rra ABa Bnarcfi nPanoPlnr).

60:120=0,5=50%' 18, r, 3x =18O; 4x= 180; r = 45.

Cyua rocrpr,rx xYtin: 45o + 45' = 90'. Cyua rynrax ryris: (45' '3)' 2 = 27O".

Bilnoei1a:90" i 270'.

640

c.,l.Li AOz

23. Ocxiar,ru r:rpaneqiro nnnbaso

l|+12+13+14=180.+180.=360..Yerrepruftryr4opinrrroe360.-240.=120". Kyr: 180'

u*o"o.

napra cyuirxnr'rx

Beprraxa;mRi rcyrn

O. AB = 8

ri4craHr aig 4enrpa

22. au=2rJ3 o" rr,

17. flpz nepenani np'fiMrx a i b yrsopunucn 4 xyrr a6o 4ni napz cyvrirxuzx: ra l4 i 24. Cyrvrixnraft 3 rrrlM

KoJ'ro B qeHrpoM

4Ji &yr\. t = 2n. Ao = zn. ali = anJ' k$.

Hacruna 2

tu-

zAOC = Bo" . 2 = 60..

rACO: dC = | dn = 4

16. E.

IIpu neperrni grox npartux yrBoprororbcff

p,axo

(cw2)'

f; 15. A.

OC I AB OC = 4 c*r.

a=4, s=o" =+=2Rz Jzz

10.

^oq

AC=AO=R=5cu.

3. E.

(cm).

pinno6e4peHrfr tAOC - (180" 60") : 2 60.. /A= ZC = = pienocroponnifi. Tomy aAOC

R2+1c2+2Rx*R2=

Cyrrixcni xyru rpaneqii r cyui cr:anoa.n'srr 180" + 5x =180; 6* = 180; * = 30' 30'' 5 = 150" - rynr'rft

13.

qacruHa

2. L.

4.8. d=aJz; 2R=ali; 5.

20. zABc =!

+R)2+(A-A)'=

6fr.

Tecr flnc caMoKoHTPorllo

rz :2=8,b

riyTa.

AC2 + AD2 + AE2 r AB2 = = (x - x\2 + (A - R)" + (x - R)2 + (A - U)' = (x = (x - x.)2 + (A + R\'+ (.r + .R)' + (y - v\2 -+ = fu - R\' + (x - R)2 + (Y + R)' + (x + R)2 = - v2' 2Ra + R2 ! x2 - zRx + R2 * y2'l 2Ry + 4R2

fpagycna mipa nnr.rcanoro rtyra AopiBuroe no.noszni oi,qnoniguoro rIeHTpaJrBHoro

C(x; R), a(R; u), E(x; -&\' Bupaa Ars cynrra xnagparis eiAcrarefi siA ro.*cu -4 go croPin r(BaApara:

= z(xz + y2) +

=!an 2

il{: TEOMETPIfl EincxiHa O.9' ra ix. Ara4eMi'{Hrft piBeHb:i::.::fi:lliti:*

Tpaneqiro Molr{Ha Blr4carrr B rco,ro, sxrr{o BoHa pirro6iuHa.

24, HexafiABCD * pirno6iuna rpaneqia, Brrrzcar{a B Aesx-e norro a pa4iycon 4 cxn, BC ll AD, An = Cp. ICAD = ZBAC sa yMoBoro, IBCA = 30o ea yuoeoro. - ZCAD = ZBCA = 30'sr rnyrpilrni pienocropoaui rpr trapareJrbHrrx fipffMr4x BC i AD i civnifi AC. Togi

A ZBAD = ICDA= 30' 2 = 60o. Ocrcinrxu cyma xyrin BCD iADC gopirnroe 180o, maeuo: IACD = 180' - (IBCA + tADC) = 180' - (30' + 60') = 90".

,:iltillliilii

TEOMETPI9 sincHiHa O. 9. ra ix. AxageraivHnil piBeHb

2l v"i rA.ri

s..

lt)

q. I

ilil

Oar

rl Tarcrau uranor,n, aACD

{acrrxa 3

KoJIa' npamoxyrnuii, a qe oBHaqae' Ep AD - AiaMerp

r i -R (pa4iycin Bnucauoro o6.rucreus,fl n.noql rpzryrHurca. Sna.rens,s

fi onucasoro ria

auafigemo e tpopuyn 4,ra

abc

S=DrTa '4R S=-

'loaiAD=2 4-8(cm)' an =L

ln = 4 (crvr), sK Karer'

ocDE: cE

- cD.sin6o"

ulo rrer(rrrb rportr r(yra 30"' flpone4eruo CE LAD'

= !!: 2

29,

=2.,6

(cvr).

Ya ABC, roAi 8D

3 rABD

Bilnoaila: 2.tE cl..

25. 3a yrvronoro BD = 2O crrn, CD = 14 cu' -+ BO + AO = AB + BO + OD ='AB + BD' r"*"= IABO = r DCO aa trepiloro ogr{ardolo (BO = CO, AO = OD 3a YMoBoro' ZAOB = IDOC stt aePrlrxalrni)'

3 pieuocti tPuxYtnuxie: eA = co = 14cm. P^oou= 14 + 20 = 34 (cu)'

t BCD\

= 9O"):

AB2 = AD2 't

BD2i

Jee

AD =

'[AB'-BD'=,t86'-Lo" AC=24.2=48(cm). S*"

Lao

t,*l'

Bucora pinuo6egpeuoro rprxyrurr-

AC.

-10X26+101

=.[to.so =4.6=24 (cvt.

AC = 1O.24 = 240 (cm').

2.240

480

AB+BC+AC 26+26+48 lo0 4.240 (cu). ]t=_ 45 AB.BC.AC 26.26.48 =33.8 p

- .ti --saE ,

DC = BC sin 60' = 3J6 '

(lD

'2 An = \ - ue4iaua,

LAC

30, V aABC ZC =9O", CD 6icexrpuca /C, AD : BD = 5 : 12. CM -weqiana, CM = 26

26. Hexair ABC - Aas:zft rpqnyrHrll(' zA= 45", lB = 60" BC = 3J6 cm. Ilponegewro CD LAB. B

O6,twcxzvto neai4ouy cropoxy-4C. BD

U {cM).

cur.

Ba snacrr4eicrro 6i€exrpzcz Hyra rpnnyruuna nona 4iaurr cropoHy Ha sigpisru, nponopqifini

.ACAD1 cTopoHaM: tsc BD

nprlJlerrtl M

flosuaqueuru xoe(piqieur nporopqifinocri l, uaemo: AC = 5x, BC = 72x, V nparroqe cepeAlua rinoreuyeu -4-8, xyrnouy rpnryrnrlxy ABC qenrp orucaroro KoJra rotty AM = CM = BM = R = 26 cm, AB = 26 . 2 = 52 (cm). Y rABC Ba reopeMoro flitparopa: AB2 - AC2 * BC'; 522 = (5x)2 + {l2x)'; 25f + L44xz = 27O4; x2 = 16; x = 4 {x > O). AC = 5 .4 = 20 (cu). BC = 72. 4 - 48 (cu). P^*"= AC + BC + AB = 2O'+ 48 + 52 = 120 (cu).

31. Hexafi

flozsa,*rttto aa r xoe@iqient nponopqiiiuocri' rolrt BD = 5x, AC = l2x'

27.

= l2o';

x'=

=5.2= I0(cvr): nO=f,en=5

cttn:

3a yuonoro

x=2(x> To.ri BD

L.s*.tzt =l2o; 2

3ox2

0).

BD = 5 cvt Ba BJracrr{BocrfiMr{ rrapaJreJrorpaMa. 3a reopenoro rocrauycie CD2 = CB2 + BD2 - 2CB . BD . cos

AC = 12 ' 2 = 24 (crvr); OC = 12 cw'

B aBOC: BC2=BO2 + OC2; BC=J|";Lr' =tE|+L44 ={OS =fa "t"' P*"r= !3 ' 4 = 52 (cv)'

28. Hexaii eucota BD 4i;rurl cropony 4C na ei'4pisru AD

i CD,

rABD.. BD

AB =

cNr,

BC = 5 cr$' 1A =

AB.sin6o"

=&f=

3 r BDC: BCz = BDz

Dc = JEc, - BD, =

^ts,- =,lri-=

642 lti; reouerptg

6O"

(cu)'

DC2 (teopeiua lli(paropa)'

EincHiHa

panufr, rpraftyrHlrr, AC = 5 cw, BC = 3 cu. -4-BC CO horo me4iana, CO = 3,5 em. ,{o6y4yelro a AC B lo raparerorpaMa, npoAoB*cr{Brxt4 CO ga roqrty O na ni4pisox OD = CO = 3,5. Poar.nsnemo rCBD, y ,sxoro CD = 2CO = 7 cm,

n4o = a (c}a).

o. 9. ra in. AxaservrivHur piBenb :i{iiii:i;r*

cos

Z('Bl)

ICBD,

cB2+BD2-cD2 9+25-49 -15 2CB.BD

2-3-5

ICBD =

12O".

Ocxi.nrru cyci4Hi xyru napaJreJrorpaMa n cymi cranoe,rsrr 180', r:o IACB = IaO" 120" = 60'.

32. V pirHo6ivHifi rpaneqii

-ABCD (BC I AD) AB = BC = 4iarona,rr,, CF = CD; BD - nucora, O - roqra ix neperrzHy. CO = 15 cm, OF = L2 cv. a BOC cq a DO.F oa ABoMa.r{yramu (ZCOB = lFOD an neprr,rxa"rrui, ICBO = IFDO st* eHyrpiuui pieuocropogHi npr traA paJreJrbrnx npsMr.rx BC i AD i ciqnift BD.

ifiiii;iiii;i

TEOMETPIC EincHixa O. 9. ra in. AxaAeuiqnrafr pieexu

l!;il

643

3 nogi6irocri rpuxyr:unrcin

coBo155 oF= oD= n=1'

Br'ItrJIr,IBae

BCD

piuHo6e4pen uit

i ICBD

CD = 5. 9 = 45 (cu)r DF =

AD gp 22

BC +

a':lll

.27 = 2187 (crsrl.

MOAvAb 2. BCryn S

2.2. Hacnlarr s arcioM crepeouerpii

Teopeua l. {epes nparry i Toqrty, ny i 4o roro nc riarxia o4ny.

He Harer$rrrt,

ifi, uoxna

3)rpra

IIr. flxuro aei piani npa- | IIn. flxu.1o 4si piani n,roI mi trarors cniar,Hy tovxy, I rqr'IH}t MarcTb crri.lsny rouI ro .{epe3 IJ14x }lo)rcHa npo- | Ky' To BoHIz treperr'IHa}orbI necru nroqllHy i 4o roro I ca no npauiii' u{o npoxo|

l Aursvepea ulorouxy.

2.16. A) (ABC) i.B) (MBC).

}rC.

2.11. B) 2 i 3. 2.

fl'arr. 2.18.f)1i4. 2.19. B, r. ' 2.17. B)

I) I\ MD.

Hecrcin.renuy

2.14.t)

rinrrcicrr.

era.

ffis*

2.15. D 1, 3. 6.

fr$ffi

2,2O, Hexafi nparrri a, b i c npoxo4nrb qepea rovxy O i neperr4uarorr, npauy d oi4noni4no B rorrxax A, B i C. Ilpnrra d i ro.rxa O (O e d aa ymororc) o4uosHatrno Brrarra-

,A)

o.qua.

2.2. n\

6egni,a.

2,4.A)

o4na.

2.5. A)

o,qria.

2.3. B) 1, 3'i

qaloTb nrtolqr{Hy cI.

* 4ani npami, qo Marorr' cni;trny rouxy A, to,4i aa axciouoro II, vepes Hux MotnHa trpoaecru nJrorurrHy, ao roro r+c ri.nrrcr'r o4uy. Ilpurryctrmo, ulo icro'rxa B' To,4i icuye Apyra trJrorqftEa' Iqo Hye Apyra cni.nrua roqxa npflMr,rx a i b npoxoAr{rb vepee npanni a i b, ane qe cyflepeqzrr, axciorri IIr. Orxce, Harne npfiuyrrleun.tr HerrpaBumne. Ilpaui a i B neper:r'rnarorbcs lTr'rfire s o,4nifi ro'{qi.

2.6,Hexaii a t b

2.7,flpunywulno, rqo rouxr A , B i C ne Jrerftarb ua ogniit npsuift. To4i uepea ,4ni npavi AB i AC Morrcua rrpoBecrrr nJrotr{rlHy, i 4o roro x ri.nsxu o4uy (arcc. IIr). Are qe cynepevurs yltoni, 3a flKoro rovxra A, B i C lerxatr y ftorr(Hifi s 4eox pianrzx npunyu{ennfi HenpaBrrJrbue. Toqrur A, B i C ;rexarr na o4nifi npauifi. nJrorqIaH. Or*ce, naure

Hexair ABC

4anrzft

rpnryruntt, BD

og-

fioro *re4ia-

lla, M neperfiHy rre4ian, rovxz A, -C i M rrew'ars - 'rorrxa y n,rorqr,rni 6. Toqna D HaJrerfi r4rr, r'patljfi AC, a sHairrrrb, i n;roquui 6. To4i sa reopeMoro 2 upnrra DM netxwrs y nlorqr.rni 6. B e DM,

ro,qi B

1 uepea np,EMy a

l::I: TEOMETPI9 EincxiHa O. 9. ra ix. AxageuivHnfi piBeHd

2.21.

rovrcy

A MorxHa npoBecrra nJrorrlr4 -

ny or. 3a arciomoro I, icirye rouxa B, r{o rre Jrerfir.rrb y n:rorqr,rni crr. Ilpore4euo qepes fipsMy a i ro'rry B nJrorqr,rny ctr. Il.norqlrnra o, i o, piaui, ToMy rrlo roqxa B n.norqrauu cx.2 He J'IelrdffTb y n.nouluui cr.

Hexaft npaua b neperlruae n,'roqtruy g s ro'rqi A To.ui A ..- cninrna rorrra nJrorqlrrru u i B, A e a'

644

lConrsa a flpsMrrx a, b i c v,ae B nroulrrHoro a 4ri cni,ruri rovKr, - roqxy O i roury neperrrHy 3 rpsMoro it (A, B a6o C). To4i aa nac.nigxorvr a axciorvr crepeoinerpii (reopeiraa 2) npami a, b, c i d tencarr nift n.norqr,rni, ro6ro ro.rxr A, B, C i O .nexar:r s o4Hiti nloqnui.

2,22. Hexafi a Ea:eta nprma, 3a asciorvroro I, icnye - JreECr{Tb rouna A, rlo rre na npamifi o. Ba reopemoro

2,8,IJexafr fiJrorqfiHr{ a i p neperunarotics no npsmift a, . b c ct.3a anciorroro II, nparaa o uicturr' yci cni.nrui

rovnu n;rorqun

nroq}r-

roura, sri ne nexa:rr na ogHift nparuifi;

2.1O. B, A, E,

2.13.

2.1

trpoBecrrr

Teopeua 2. ,flHrTIo 4ni rovxa npauoi na;rexarr, u.norqraHi, ro i Bcs np,fiMa HaJrexrrr qifi nnoulnni. Teopeua 3, 9epea rpfi Torrnrr, rqo Ee Ha,'retfiarr uprrvriii, Mo)+cHa npoBecrrr nJrorquny i go roro :rc ri.nrxz o4ny. Ilpra poar'aeaxni crepeouer:pfiqnrzx sa.qaq fleprn oa nce neo6xigno cnpo6yoarz

4)4si napa;re;rr,Hi upaui.

uiii naorqilni. i roqmfi, srti He sarexars ifi.Ac d,C e o., l x ri,rtHu o4Hy. HaJre?+(arb

neperutarorbcrr tro npfi-

nu4i.nuru trJrorqrrHy, rraM caMrrM eniuur,r sa4avy 4o n.nanitrerpuvnoi. Il.norqr,ruy oAnoaHarruo Br{aHat{arorb: 1),qni npsui, rqo fieperr{rrarorbcfi; 2)nparra i rovra, rrlo He HaJrextratr qifi rparaifi;

crEPEoMETPli

2,1 . Ocxoeni noxqrrn crepeoMerpii. Axciorvlu crepeovrerpii Axciomz crepeouerpii: I.. flxa 6 He 6yna nrorrtrrn]a, icnyroro rorrxu, u{o

rrnrs i npauifi c neperaHy l,rorqlrn o i

#=#=i

9 = 36 (crvr). 3a yuonoro BC = CD = 45 ctn. Ocsint'wuABCD * pinno6ivHa rpaneqia, ro AD = 2. FD + KF = 2FD + BC = 2' 36+ 45 = 117 (cu).

Torrxa EaJrex{r{rr ycim.rpr,ou nnorqr,rnau cr, p i y, a ana.rrzrr,, aa arciouoro II, rourca C laarre-

ICDB, toil

roAi CD = 5x' DF = 4x. 3 aCDF Ba reopeMoro fliiparopa CD' = CF' * FDz; 25x2 = l6x2 + 729; 9x'z = 729; flognaqunro noeQirlienr nponopqiiiuocti

x2=8t:x=9(x>O).

n.norqi,rurz

b, nprauouy - no npmrifi npaui a i b neperr.trarcrscn n rovqi C, ro qn

6icexrpuca ZCDF. Sa sracrneicro 6icexrpucrz

2.9.frrqo

nrifi a, n.rrorqtnr 0 i y

Ocxi,rrxv 3a yMoBoto BC = CD, to

nponopqifinicrr' rignonigrrax croplH:

!tt{li:{:it*

itlililii;:t

IEOMETPI9 SincxiHa O. 9. ra in. Axaseuirnrafi pieenr

tj:i:f 645

r 2.23. la:ai rpffMa a i ro.rxaA

Bu3naqarorb nlonqlnry

(te-

operra 1). flrcrqo npaua b npoxogurr, vepea rovxyA i neperuuae npsMy a n roqrli B, ro nparvra b Mae s nllotrIunoro cr

4ni piani cniarsi rovnu (A

rqzri

i B), rouy

Jler+curr

y nro-

(reopeua 2).

2,24, fl.xr4o ro.rxu A, B, C i D He rlexcarb a oAHift n.norquxi, ro sci sour,I i l*cogui rpu B Hrrx He Jrelr{arr na o4nift npnuift, Tomy xoxua B qorr,Ipbox MorrtJrr{Bl,rx rpifior rouor (A, B, C\, (A, B, D), (A, C, D) i (8, C, D) ozaravae eAEHy rJrorr{uuy (teopeua 3). Ifi vorzptr nroqtrHr e pisnrnmlr, 6o arrqo 6 6yAr-sni Asi s sux a6irarucs, ro rouxra A, B, C i D Jre)fia;ru 6 n o,qnifi n:rorquni, rqo cynepevurr yuoni.

2.33.8)

nepcmuHg naou4un).

flpaua, no srcifi peorrerpii: 1. ,fxrqo gri

rorr, nJ'rorqzHy o, Ocrimxr,t npsui AC i BD .ne:rcarr y.n.lroquni c, ro i roqxr A, B, C i D ;rerxarr y qifi nnorquni.

I I

2.27.IIponegeuo npsrvri AB i AC.3a arciorvroro II, nporeAeMo rrepea Hrrx fiJrorqilHy o. 3a yMoBoro M e AB, K e AC, ro4i rosnu M i K .nen<arr y n,rou4uni cr, a sHaqr4rb, i npana MK aer'wru y n.rorqnui o (TeopeMa 2). X e MK,

Torrnr{, To trpffMa, npoBeAeua vepes

AaH[x nJrorqrrH.

2,34. Toqxa neperuny X nparvror KI i n.roll1'uau (ADC) qe roqr(a neperr,rny npamoi ,I{F npauoi DC (DC c ADC);

X cni.nrsa rorrxa firorquH - ro BoHa rrexurl Ha (BCD) i (ADC), upfiltifi reperzry qux n,rorqzu DC. ocxi.nrxr.r

2.35. EF c (C\CD), nxotlnuu (C.CD) i (ABC) neperuuarorbc.fl no npsuift CD. Or*ce, uryrana rorlrda r{e roqra neperrany 1{ npsMr4x EF

-i CD.

X e (!.

pom6, a ll BD.Ilapa;ren xi npnMi a i BD oAHOsHaqHO BrrSEaqaroTb trJrOrlUEy C[, TO'rKa neperr.rHy gia-

2.28. ABCD

rona.neft pou6a O rraJreJrc[Tb uparuii BD, a or:rce, i nrou4vui o,. Togi giarona,rr /.C poilr6a rraJrexrrrrr, nnorqr.rui o

(reopeua 2). Taxlru vr.rxou, yci BeprxrdHri pou6a zr Jrerfiarb e oaHir?

naouruni.

i upaua

ry,

,{ore4emo MeroAoM ni4 cynporueHoro, rqo npaui CD i a neperunaror6cs. flpianycrr.rMo, rr{o BoHr,r He fleperrruarorrca. To4i CD ll o, i vepes rouHy D, rqo ne Jrextllrr' rra npamifi a, npoxoar,rrb 4ei npaui (BD i CD), napa,re.nsni c, rrro cytrepequrr axcioui V (axcionri napa,rerbnr4x ),

2.36. PQ c (SBC), rosi X e BC.

2,37.To,rxu D

nextart

e n.norqr.rni

(AMD), i n nrorquni (CDB), oruce, DM fi:r.ia [eperr.rny rlux nJrouI{H.

Bftcxosox: npaui CD i a reperuHaromca.

2.3O. B) (B,BD);

646

I) (/rD1);

A) (,438,).

3, 4.

lt{: TEOMETPIq Eincxixa O. 9. ra ix. AraAeuri.{Hufi piBeHb :;:iltii::iiiil

$2.3. flepepisr 2.31 . B)

2. fxrqo 4ei napaae.nrni r,rorqrana uepernyri Tperboro, ro npami neperuHy na_ pa,re;rrHi. cyrr merogy cnigir noanrae n no6ygoni 4onoui'Icnoi fipfiMoi, axa e ao6parxennarvr .iriHii nepernxy civHol n.norqunr4 s nJ,rotrIr4noro rpani $irypu. v sara.nrnouy Br4rraAlcy ciuna n,ro'4una leperrruae [Jrou-rr4Hy xoxnoi rpaHi nrnor_o,rpaHrrr{Ka i ronrry 3 rrp,fiMax, rra fircux Jrertarr, pe6pa mrrororpannxxa, II{o6 no6y4ynarn c.ni4r.r Ha rpansx i pe6pax, oorpi6"o, I ) no6y4ysaru roqru [eperr4Hy caigy ,pu"",r"; " 2) gni rburn nepepisy, sxi rerxarr r ognifi rpaai, e,e4naru.

ua o4nifi nprrvrifi, qepeo ra}tx Mo}Krra fipoBecrrz 6ea,ri.{

t^:x:

l,roqzulr uarorr 4ni cniltsi

TOqKr.r, e npfiMoro nepeTr,rHy

nJronltrH. gauuir uorupurcyrxr{n, giarona-ni axo2,26. Hexair ABC D ro AC i BD neperanarorrca n rovqi O. 3a axciouoro II, npnui AC i BD, rqo ueperrnarorbcs, oAnoqacHo BI4BHaga-

rpaui, Hasrnaerbcs cri-

flo6y4ona repepiaie Muororpangunir ipynryerrcr ga nac;riaxax s axcioM cre-

MOlrcHa npoBecTrr eAr{Hy rrJrorqrrHy.

Jre}rcarb

fiJ'roqr.rHa nepepiay neperHHac nJrour,rHy

dom ntouqunu nepepi,ty.

2,25, a)V npocropi icnye 6eariu rovox. ui4miuunx ri4 4anoi ro.Jlcl,r.A, Biasmeuo tovrcy B. gepea roqrcrl A i B npoae4ervro npamy. 3a axciorrtoro I, eisrveuo e npocropi rouxy C, ara He Harrexcrrrb npamifi .43. 9epea nparry AB i rourcy C uoxcsa [poBecrl4 eArrHy nJrorqraHy. Ocri.rirru i ro.*cy B, i roqxy C rvrolrtna sa.nrra 6eariqqro cnoco6in, ro i nlorqun vepea rovry A Mo)fiHa flpoBecrn 6eaJriq. 6) 9epes Ani Aani roqrcz -4 i B npone4euo eArrHy rp,ffMy AB (axc. Ir). 3a axciolroro I, icnye 6eeri.{ Toqox y npocropi, gxi ne na.nexarr' nparvrifi -A3. 9epea rcor+*ry rarcy rouxy i nparvry-AB Ba reopeMoro 1 rvrorxsa trpoBecr[ nJrolqr4Hy. Orr+ce, vepea gri r:ovKil MorfiHa npoBecrn 6eaaiu n.roullrx. n) ,flrrqo 4ani rpra roqxn He Jrer{arrr na o4xifi np'nuifi, To sa reopeMoro 3 uepea uux

ornre,

'flxrqo naorqzna rpani Mrroronpa'Hrr*a i n,rorqzna nepepiay rvrarorr ,rsi crinrsi 'roqftr{, To BoHrr fieperuHarorbc.a no npamifi, qo npoxoAr4Tb vepea r1i ro\nu (niHia

Bidnoaidu: 4 n.nonlunr.

flxrqo xc rpr{ Toqrr,r

iti:::jitii$l: TEOMETP|9 EincHira O. 9- ra in. Axa4eruirxrai piaexu

{J:l:

642

y nipauigi ABCD roqxa M roqxa reperlrfiy rregian rpani rASC. flnou{nna nepepiay, u{o npoxo-

2"39,

2.38. fl;rorqranu (ABBr) i (BCC,) ltarorr.+si cniru-

Arrrr' qepea pe6po DC i ro'rxy M, neperunae flJrolquD uy (ABC) no npsuiit CM. K - roqrca neperr.rny npauoi /N

si roqxu B i 81, roMY ne- myraaa.ninis peTrrEy qr4x nJrorqr,rH. C, B.

BB,

2. 44, TIo6y ay emo nepepia xy 6 a ABC DA rB rC rD, n no rr{uHoro tlc, fiKa rrpoxoAarr, qepeg Bepurr.rHy B, i tovxnMiN,geMeAAr, Ne CCr. Ocri,ruxr ro.rfir4 ,8r i N - cnilsHi .4JTff n,tou{Hrr L{

i (BB.C\), ro nparra BrN

Ivrac rsi cni.mni

CM i nnoqann (ABD)-

a-7 "'77"''," |IV 1/

nluni

npruiii

3',

ALKB

a (da,n; -- AL,

(ABO\ uyrcaHuft nePePie' c1

a i (AArDr), iL - trrrorq.un ai(DDrCr). 3'e4naemo sigpiexaMra M i K, K i L, L i M. BILKM ruynauufi nepepia.

ttlo BrrxoAsrr' a eepurunu Dt' Iloagaqurvro cirrHy nrorlr4Hy {ALC p). (ArCrD) n (A,B,C,) = A,C,; (A,C,D) a (AA,D\ = A,D;

Morxe rpanzrucb, ntro ror{Hrr M i N posrvlirqeni rarcru r{unoM, rrlo rpnrvra XY rre rreperuHae pe6pa Hnxnr,oi ocnorz ry6a. V qronry B[naAKy moxua snaitrz rovxy fieperr,rHy

Z npawoiXY

a npflMorc CD a6o

nolygyearu

NK 'l BtM IMK ll B,N).

I; pl6"p,

2.45. no6y.ayenro nepepis niparui-

pn ABCD rrJrotqrrHoro, qo rrpoxoAZrE,qepeo rovnu I{, S i P, .qe K e AD, S e AC, P e (ABD). IJo-

(App)^{ccp)=DCf' aArCrD

nJrorrlr4n.

rrJrorr{EH

ga ABCDATBTCTDT n.noulrnoro' ffKa rrpoxoAr'rrb qepec

,'-

qux

treperr{Hy

2.4l.Ilo6y,qyerronepepianpsMoryruoronapaae'nenineC1,-

(ABC), roMy Mae rre?r(arrr Ha

.4-B

Ana;rori.ruo anafi4euo c.ni4 Y npsrroi BrN na nlorquri nzlxnroi ocnonu xy6a. To.rrlr X i Y -: cni.nsni rorrxrr nJlorquu u i (ABC), roMy flpsMa XY e npauoro nepeTh Hy rlrrx n,lou{ff H. XY nAD = K; XY aCD-- L. M i K cni,rrni toqrcn D

(BB,C') = BK.

Hiuqi At,

(AAP) i

- npnmifi AB^MBt=X.

1) flaorqrzHa (AbO) neperr'rnae nJ'ron{rruy (APpr) no rpanifi KL, KL lL A3, ocrcixr,su AB 1l (ArBrCt) (ArBrCr).

- npfiMa rreperrruy nJrotquu tx Bnaft4emo caiA npsnoi BrM na nronlrmnroi ocnonr ry6a. Ide cni.nsxa rovHa

rrrrorqrrH

V rty6i ABC DArB ,C ,D , r - O -- rourca neperruy Araroga-nefi rparri A,B'C'D'. Ciuna rurouluta (ABO)'

2\ (ABO)

4ux

i (AAP\),

2 .4O ,

i KL c

B, i N.

Anarori.{ao BrM

e np,fiMoro neperrzny

trJrou{}ru. 3'e4naeilro rorrHra

//f \ Ha nroulr.rHa / Illtt \ roqxw D i K o rpannro -ABD, d( ll/lhA .) n roai DK - riniq ilepernHv \ lll-lrut r/ civnoi n.nonquu u i rpaui ABD. \[l'/'aDCK - u:Yxanuft nePePio. Y CiT

utYnauuk nepepis'

BHaqnMo

ii d. tx .

(ACD) = 175.

ToqHzl{i P.irexcarr y (/BD). IIpone4emo npauy /{P , KP a AB = L.

2.42.Hepea rouny B i nparay /D Morrcna npoBecrr [irorq[uy (noana.rrzuo ii a) i ,qo roro x ri.nr,Hrjr oguy' 3nafigeuo, sK rl,rr rrJroilIl4Ha rleperrrHae rpani i pe6pa ry6a' Ocrinrrcu roqxu A i B aelrtarr y unorqrzni cx, ro

' Toqru S i

y ({BC). Kl,S - uy-

.L ,remarr

TTpoeeremo SL. a xaNnit nepepie.

ABca.

fl.noulunlr (AAID \) i (BB p J ndpa:re,rrHi, roMy rr'roq?rna o fieperr,IHac ix no napaleasHr'IX [psMr{x: BCr I ADr Illynautii nepepis ABC.DT

V 4pyrorrny BlrnaAr{y ro'rxa P ua

(/3D)

2.43. qepee,qiarona,u -4D, rpani AA.D TD i cepe4zny M pa 8.B, uoxua npoBecrrz fiJrorllrduy o4ny. floaHaurzvo ii q..

pe6-

4o roro :x rilsxra

Ocrcinrxu ro.rrra A i M e cui;rrur{Mrr AJr'fi rrirolqug cl i (-4BBr), ro AM * rrpfiMa rreperrHy 4I4x rrrorquu' 3'eguaervro ro.rxNA i M silpiaxoNr.

fl.norqunu (AAp) i (BB1C') napa,te.nsni, roMy nrrorqrrua qepea rovny ct neperr.rHaâ&#x201A;Ź ix no napa.ne.nbHrax rp.fiMr4x. M napale.;ruro npa*rifi,4-D, [poBeAeMo nprIlry MN (N roqfia [eperrrny o pe6porvr BrCr). cni.nrui toqnpr rrJlorqr4H a i (ArBrCr)' OcHi.nrxn ro.rrrra N i D, nep-er:rany

qux

nJrotr{r4

Ir.

n.norqurri

posuirqerra raftnM qfiHoM, r4o flpfiMa KP i nigpieox 1B He neperunarorr,cx, To4i suaxoAr4Mo cttil, X Toqrty neperr,rHy npsmux KP iAB..KP = X, KP BD = R. ^AB flponogr.rmo npauy SX: SX BC =^ @. r{orr,r^ nepepie. pr,rxyrnlrx KRQS * uryr<auzft

TecrAnc caMoKoHTponlo r{acrrana

to NDr

uprma

Ana.rorivuo, AM * tpawa fieperr'ruy [JroqlIH o'1(ABB.)' rrorrlpur(yrrrvo AMNDT. Illyxaurafi nepepia

2.8.

1. B.

4. B.

6. E.

7. A.

9.A. - 10. B.

64a

:#!: rEoMETPlg Einsnixa

o. 9. ra ix' AxagervrivHr'ri

piBeHb

i!:#:itiiil:i

]lilfiiiili!.r fEOMETPI9 Einsxixa O. 9. ra ix..Ara4ervri,rnrai pieexu filli: 649

ABC i ro.{ra ney nrouluni a. AMr, BMr, ftoro CM, .._ megianu rpr{rcyrHrrxa, AII r BH z, CH

21. [si Bepmrnu HanpuH.;ra4,

.perrrrry lregiar

cepeAssr ttiuis rABD.

MN =;DB.

Brrcorr.r,

M e u, M" -€

Ocxi.nrxn rpan.fiMrr xy6a e pinni HBaAparu, ixni AiarorraJri rar(ofir pieni, touy MN = KN = MK.

-A

i B rplrxyrnlatta

Jrelrcarb

roqrca fieperrrHy Brrcor. MMrc a, Q e

MM*

cr, roMy

3ri4cu ezrnzsae, qo BC - d", AC c d". Toqnu A i fI, .ne;xarr y n.norqzni u. H e

cuHea. 12.

13.

f, .U.

14. B.

e a.

AE' sri[-

22, (xrxrx"), (xrx"xn), (xrxrxn), (xrx"x). 11. E.

23. lue. !& 2.24.

24.

15.

2s 16. B, f.

|iacruaa2 qo npflMa 4 ,{onegemo qe rBepAxeuHfl Mero,4oM ni4 cynporznnoro. flpznycrumo' neperrrHae nci croponra rpunyrurxayrouraxK(7(e AB)' L(Le BC)iM (M e AC)' ocrci.nsru TpfiKyTHrrn JreE(r,rTb y ,4eanift n.norquni cx,, MorI(eMo sacTocyBaTu anciomy BSaeMHOrO poSMlrrIeHHfi TOqOTC Ha nJroil{I',rHr.

17.

flpaua a neperrllrae ni4pieon AB, roiuy

roqrcrz

AraroriuHo, roqxn C i B nentarr y piaulax nlBrrJrou{nHax. Ale, bcnilrxu npsMa a nepernuae ni4piaox AC, To roqftrr A i C raxox Jrelxar-s y piaurx niennoulunax aigHocno npannoi a. IIe osnaqae' rrlo roqxa C o4novacHo Mae SHaxoAurr'Icfi B ABox piauux ni elroulnuax siAuocso npxlroi a' rqo He-

K tA

M C

Ilparwa a He Mo)r{e neperuHarr't rperro cropouy rprrnyrrruna. rpra npami b, c i d tpoxoq.flrb qepea rouxy ? i nonignoyroqfiax P,QiR. 9epea rpalry a i Torrxy ? moxHa npoBecrr'I fiJrorquuy (nocuavumo ii a) i ao roro lr ri.nsmn orsy. To.rxra P i ? npamoi b Ha,telxarr nJronIXHi oc, ro4i aa reopeMolo 2 PT c d. Ana,rori'{Ho TQ c ct, TR c a. P € a, Re oc, ro4i PR c ct. Q e PR' rouy @ e g.

18. Hexafi

?, P, Q i R

Jrelnarb

r ognifi

MilsP

Haue npznyrqeHH.a He[paBrlJr]'He.

Bracnoeoft: rourcrl

A1 -w^r:.u''";.

A i B ;rexarr y pianux ninnrorqil-

Hax signocro A.

Molr{JrqBo.

nepeTI4HaIoTL nP,flMY 4 BrA-

n.nouluHi.

19. Hi.

'' 29. Hexafi npnui a i npoxoAr,rrb qepes

9acruna 3

rovqi O, Ilpama c He roqfiy O i neperzuae npami a i b riAuoniab neperranarorr,ca n

nosroqxaxMiN. Ilpxui a i b aa arciouoro II, ognoanavno BusHaqarorb n;rotrI[ny cr. Touxn M i N npauoi c Harexarb n:roulnui d,, a or,fie, i npnma c Ha-flexctrrr, n.rrorqtt ni cr, 3O. Hexafi Aec.rrrb fipffMrrx fipoxoAflTb qepeo Toqriy O i neperuHarorb trpsMy a. r{epea np,fiMy a i rovxy O uontHa trpoBecrz nroulr.rHy o, i 4o r:oro rr ri:nnu oAny (reoperyra 1). Koxcsa a Aecsr[ trpsMr.rx Mae B rrJ'rorlrrHoro o 4ri cni;rrni roqrcr.r: O i rouxy neperlHy B npflMoro a. ToAi aa reopeMoro 2 xorxsa B qux trpsMzx rex[Tb y uorquui ct.

31. Yripani4rKLMN P e LK, Re 1{N, S e ML.

20, rlepea npami

-A-B

i ,4C, ulo neperr'rnalorbcfi u tovqi

A, uorxH€r nponecrll nJrorqr4ny (.4-BC) i go roro xt ri.nr,rcrr oAHy. Toqrru M i N HaJre)fiarr, ni4noni4no flpfiMrrM AB i AC, a or)fie, rerrtarb y nlouluui cr. BracnoeoH: rouxu M, N, B i C Jrextarb n oguifi n;roqrzni.

Ilo6yAyeno nepepis nipauigu uoqzHoro (P/?S).

l) P e (LKM), S e (LKM), roruy PS c {LKM), PS=(PRs) a(LKM). 2) P e (LKN), R e (LKN), romy P-R c (LKN),

P1'r=(PfiS)^(IKl/).

Bnaii4eno 4ni croponn (pirypn repepiay: eigpiexz PS i P.R. Toqxa S cninrHa roqfia ABox nr'roilInu (PRS) i (rNM).

650

r:;::::

fEOMETPIS EincHixa O. 9. ra ix. Axa,4euivHhfi piBeHb l!i;i::;:+i;ii:

ili::ili*iil:

TEOMETPI9 EincHiHa O. 9. ra ix. Axaseuivxrafi pieenu

f:;1 651

r $

a MBCN - rpaneqis' - naparre,rorpaM, rrlo Marorr' cninrny cropoHy BC. BC ll MN sx ocnosu

3.18, ABCD

rpaneqii, BC ll AD JrorpaMa. "lop.i

.alc nporr4Jrelr(Hi

MN ll AD

3.2. BsaeruHe po3Miu4eHHc npeMoT i nnotr1nH14 y npocropi Moxnuei raxi nzna4nz

croponz napaJle-

[pflMoii i nraorrlaur4 y

BBaeMHoro poeuirqenna

sa o3narcorc napaae.nrnocti

npflMnx y npocropi. M rpaneqii4 A-BCD i PZCD' - cni.rrna ocnoBa srti He rexcarr e ognifi r.norqusi. @.R - cepegus linia rpaneqir ABCD; QR ll CD. Asa.;rori'mo, MN ll CDToai Qli ll IWt'l sa oaHaxoro napare,rr'nocri npflMnx y npocropi.

3,19. CD

cr i npaua a napa"re.nsui a ll cx, (Heuae cni.rrnrzx rovorc)

Ilpaua I reperrruae

Ilpama rtt Jrexturb y n;rorquni cx

Ilnorqraira

r rouqi A lr:a=A

nJrorqrrHy cr

nca

(Bes,riv cni;nHux rovox)

(O4Ha cni.nrna r:ovxa)

Teopeua 3. ,fxrqo oAHa B ABox napaJreJrr,Hux rrpfiMfix nepeTr.rEae nJrorr{r4rry, To i 4pyra npaua rar{ox nepertrHaâ&#x201A;Ź qro nJrotrI[Hv, rpuldyrua nipami4a. Touru I{, Z' M, N SA, AC, BC, SB niAnoni4Ho. sA = sB = sc = b, AB = BC - AC = a.

3.2O. SABC cepeArzHr.r

-ni4piaxin

V ar4SC KZ

cepegua

"Ioti KL = 1cs = 9,

V IABC ZM

i I

!:ii./-.:;:::.i il+!;it:+:.:.i

$ii1:.*':J !ildil*::::: i;Ftil1;!:.:::;i

ilt$*-l:::::.:l

3.22. [). 3.23. B) CD;t)AB;

(ea oanavennau).

anaroriquo

cepegna nivia,

AHanorivHo, KN =

l,rY:r..-..'....

rinia

BC;E\AD. 3.25- B) BiApiaoH.

2 r 4. fl.norquna o

Ocxi.nrrcz rrps:o.tiAB, AC tAD neHa o4Hifi n.nor4zHi, To ro.{'r{arb xu Br, C, i D, cnirrHi roqrrz

.rirrlrr, npocrip Ha 4na ninnpocropz.,flnrqo roqxuA i B;rexars y piaHzx

i NM = Lcs -9 22

[)

3.24.8\

ninnpocropax, ro niApisox.A.B fieperlrHae nJrorr{rrHy cx.. Hannaxz, axrqo rourz C i D.nerxarr, B oaHoMy nisnpocropi si4uocHo cr, ro nigpiaox CD ne neperunae rlro nJrorqlrHy.

=!tn= , 2

!ds=9.

trJrorrlrH (ABC) i o. Ornre, nonu relrcarr, ira cninrnifi npauifi qux rrJrorqrrH i yrnoprororr niapiaox.

bbaa 2222

P*"r, =-+-+-+--a+0, 3.21. Toqrcz A, B, C i D ne ,rerxarr, 6 oAniji naorquui. cepeArrltu ni4piaide AD' BD, To,txu M, N, K i Z

MZ cepeAHs niHis a.A-DC: 3a ossase*:r.flw MZ t DC.-2lwt,=-l oc. BC i AC,

AB.

Ana,rorivno, NK

3.26.3a

cepe4xa .irini,s aBDC:

, B.c. = ABl B'C' Bi1noeiOa: f) 6 cn. "Ioni

MZ ll NK i MZ = NK.

flonroprenrra nipxynanna, e rprzxyrguxin ABD t A-BC MN I AB t KZ ll AB, sr.ilv.u 1

MNllKZ. MN=ZK---AB. 2 Are aa yuonoro CD = AB' toti MN = NK = KZ = MZ. gorupuryrnux MNKZ poiu6 (nporra;rerxHi cropoHu napalellui, rci pinni). MK I NZ nr giarona;ri pou6a.

3.27 , 3a oBHaqeHHsM cropoHu

"toai KM

o*,,

=IUU,

;::# rEOMETPIq Einsxixa O. 9. ra in. AragevrivHrfi piBeHb iii::;i;ii;iiiiii:i

A tOB

rwN

+ B,c, + ,+,c1 =

|r*

B tOC

=!a,c,,

Lo,", )",c,.1r,\c,

Bilnoeidu:

KM, MN i I{N

pelHi ainii rpuxyruuxin

- f,,+u,

654

rpzr{yrnuNaABC treper[-

rpfluifi BrC,. BC ll B p r BC = t5 cm, AB, : B rB = 2 : 3. yABC * aABrCr 3a ABoMa xliramr.r (ZA cnin:ruuir., ZABC = = /AB:C, ax ei4noni4Hi trpu trapaJreJrsHrx npsMrx BC i BtCr i civuifi -4-B). AB1 'BC 2 .15 M BC AB, BC ("*)'

Nx =Loc. NK I DC. 2 ToAi sa ogHaxoro

yirronoro n.nonluua cr i nJrorq[Ha

Harorbcs tro

ABr + BB1

= s =tt

nignoniguo ce-

t i A.OC

=;4c,

",r.

=: +* 'M,B,c,

ii:;iif;:lltit TEOMETPI9 Eincxixa O. 9. ra ix. Axa.4erri.rxrafi pieenu

iil

655

6y.xo 4one4euo, rqo sci npaui, napa.re;t'ui ognifi a 4rox npnnczx' sxi rreperr{Harorr,c,fi, rqo [eperrrEaror], Apyry np,flMy' Jre]fiarb s oAHift naorqzui' Orrxe' nparui-4Ar, BBliCcrJrelrcarb s o,qnift n;roulrani. Bnauzrl, 'ro'tt+uAr,8,, C'nexatr

3.28, Pasirue

na nparrifi ArB, neperrzny nJrorr{t4H (ABB1) i o' ri4piexaArBr' A qe oenavae, u1o CC, 3a reopemoro'@-a.n eca C, - cepe4una n,s ;rini.n rpaneqii -4'ABB, aa reopeuoro rrpo cepeAulo niniro tpane4ii:

=!.(AA o' Bi0noaidr': f)

cepe4-

croponrE Tpr{r(yrnllfia

i napa:realna rperifi cropoui'

po'r4 uxx rpurcyrunxin noAi6si:

BC AD,

""1

n.notqr,rnu o.

BXC (BC 1l AD aa yuonoro).

4*,t #= 8 4,5

8. 4,5 (AD, - D131 .1.5 + l,5

Bi}noeidt:AX=4cw.

8.4.5

3.35. Il.norqrana (ABC) np,fiMonyrHr,rxa ABC D nepermtae trJronlrrHy cr no npauifi AD. Togl aci cniarxi rouxz, n rouy vr.rc.ni fi ro.rxa X, .nencarr, na npauifi.AD.

Bidnoaida: A) 6 cm.

gatnit' niApisox, av'uir se [eperr{rrae rrJrorlrnxy a, flpone.nn MM, ll KKt ll NNr, Kfr = 9 cru, NNr = 15 cu' Paninre 6ylo goxe4eno, rqo npaui MMr KKr i Nll, nexarr n o4niii nlolluni' 3a reopemoro (Da,reca K, - ce' ninia rpane4ii peAzna si"qpiaft a M,N r, a KK, - cepeAu,fi MIMNNL1MM,1l NN,):

3.3O, MN

!ir;,,.

-....

{i;i'ri:l{.itr

liiif;::i,;::.li

K * fioro

3.36. Ilo6ynona orucaHa s sa,4a.ri 3.35. 3a ymonoro SM : MR -- 2 : 3. Togi, ocni.nrxr MN ll RT, ro SN : NT = 2 :

yy, = l1MM, + NN,), sP.is$u 2KK,= MMr't NNi ' 2' MM,= 2111, - NNr = 2'9 - 15 = 18 - 15 = 3 (clt)' A) 3 clr'

3'31. Hexafi P e u, Z E 0. .IIpoBeIfi PK ll ZM' PK = 6 nparrl PK rr{r4ny c, no

roti

i LM

ctfl,

ZM ='9 cu. flapalre.nrui

F, ,srca fiepeTl4Hae nJIOnparrift PZ. Baywonow ni4pisox KM neperunae nJronlr'IHy cr n :ro'rqi O'

e PZ. * yOZM

oguoa]flauHo Bt4sHarraroTb AeJrrdy nJtonII4Hy

PK OK . MZ = OM -Hexafi OK = r cM, ro,qi OM = MK - OK = (6 - r) cm. 9= ", 6(6-x)=9x; 36 - 6x=9x;irlx=36; x=2,4. 9 6-* OK = 2,4 cv, MO = 6 * 2,4 = 3'6 (cu).

I PK i ciqniri MK):

flpaua i

rrJrorqr.rua MoJ{cyrr,

6yru posuiqeui

"r-i--1

--T-a--r r

upocropi rardr4M trr4roM:

a) rpsva Jlenurb y naoqrani; 6) npama fleperrrHae rrJrorqusy; n) npnua napare,rbHa n.norqrzni.

656

Bi1noei1t:N?=6cu.

roi-4rC, tCrBr=22$. To4iA.B, : ArCr= (2 + 3) : 2 = 5 : 2.

3.32. a)

patazir ni4piaox. C e AB,AC : CB = 2:3. Ocxi.nncu napa.ne.rrni npnui,44r, B B r i CC r rreperr.raarorr, ri4piaor AB, ro sci BoHE re:rcarb n o4nifi n.norquni, ara neperrrnae [Jrorq[Hy o uo nparriii ArBr. 3a reopervroro @a;reca, ocxi.nrxz AC : CB = 2 : 3,

aa ABoMa

N .R

Ba yaaraJrbrreuoro reopeMoro <Dareca. ,SZ = 10 cu. "SN 2x, NT 3x, ST 5 = = = ti 5x ='l.D; x = 2. sN= 2.2=4 (cm); NT= 3 2=6 (cm).

3-37.HexaftAB

liyraMr (ZPOK = IMOZ sx rep'rraxa;rrni, lK = lM sx rnyrpitrni pianoiroponni npu yOPK

DM : MC = 2 : 3'. To,qi 6a y3araJrrtHeHoro reopeMoro (Da.neca DX t XA= 2 : 3. Orxe, ro.rxa X Mae 3rraxoAr4Tr,rcr na ri4pisxy AD rax,.iqo6 nraxonyBa;racb qfi pinnicrr. 3a y*ronoro

cepe4r'rxa'

Bidnoaida:

^ADAX '' BC BX Ijexafi AX = r cM, roAi BX = AX + AB = (x + 2) cu. qe '= 2(t + 2\ = 3x; 2x . 4 = 3x1. x = 4. 3 = -:-; x+2

IABC a aADrD, a orlrce, ri4nori4ni cro-

Maeruo

3.34. IAXD

ei4rnnae ni4 nroro rpr{nyrHun, no4t6nuit AaHoMy.

3.29,

pernuy HerrapaJreJrbHrlx rrpfiM{x CD i AB. Ocri.nrru npama CD q'u,, ro X roqxa neperprny npauoi CD

14 cu.

Ilpsna, fifta treperunae gni

3.33. Ilapaaersiri npaiui BC i AD oAuooaaqrro Blrorraqarorb rrrrorrluay (ABC), axa neperlruae nnoqr.rny

+,88,) = I .1^2+tG\= 14 (cM).

ac.

rrJron{r{H (ABC\ i o Jrerrtarl sa ix cnimnifi npsrvrift -4-B (axcioua IIr). Ocxfumxn X, e (ABC) i X, e u, ro X, e AB. ,{;rr no6y4onn ruynanoi rouxr,r X, na nigpiary AC (prc. a) a6o ua iioro npo4on:rcenni nponeHi AC (pzc. 6) siarMervo rouxy X i npoeegemo nigpisox XX, (X, e AB).

Vci cnimni rorrxr4

3.38. Ocri.nrrcu nparriAB i BC neperuxarorr, rrJrorqr4rry c, ro BoHn ifi ue napa-nenrni. AI3 ll DC, BC ll AD sft flport4Jre]fiHi croponlr naparer'rorpaMa. ToAi ra reoperrorc 3 (rro4y;m 3), nxqo npaua AB ueperarrae nJrorqrray cx,, ro i nparua DC ii nepernHae.

'-'/

:i;i: fEOMETPIS Elncxixa O. 9. ra ix. AxagervrisHrfi piBeHb

i{tiii*ff

;ii:illtl:.i;;{'! FEOMETPI9 EincxiHa O. 9. ra ix. Axa4euiqnlrft piaens

ii:j

65V

.lew"yrrs y nlorquHi a, a tABD ln - ix cnilsna cropoua' rqo HaJrelfir4Tb neperrrHae CD npnuifi neperuuy fi,rloqrrn i B. Ilpaua r,roqnHr cr i F siAnosiAHo B roqrcax C i D. Eygr-nxa

3.39. Hexaii yABC nrouluui

a Si?@

cx'

o a PfiT

npnivra, trapaJreJrbHa CD, neperzuae nJrorrluHr4 B aa reopeMoro 3 (rrno4Ym 3).

(aa 4noua

osxa.reHss. flpaua naaasaerybctr naparNenaaoa nnotu,urti, JlHrrlo BoHa

He Mae ,fioA-

uoi cri.nrnoi Toqrcr4 3 Hero. ossaxa. flxrqo npaua, sr{a He HaJrelfir{rr nnoulnHi, [apa.'IeJlr,Ha nrcifi-ne6yAr npsuiji ga uifi nnorrllrni, To BoHa lapaJrerr'Ha i cauifi l,noquHi. Hac:rigorc 3. flxrqo np,sMa fiapareJrbHa Hoxnifi ia n.rorqzn, ,flHi neperfiHarorbcff, To BoHa [apa-

Hasriaon 2. Icnye 6ea,riv

lapuerbrra rronlzni, ro trpsMrrx, RHi napa-ne.nrHi qepea rco)lcuy rovxy qiei o4Hiii i rift cauifi nnoIIJTOTqI{HII MO)fiHa npoBec-

rrrr,rni.

rH e sifi flpsMy, [apa-

ae.nr,na

ne;nny 4aHifi.

T14Hy.

ry, 3.4O.

BD, ro aAOC N a BOD

TosiCO:OD=AO:OB=3:5.

npcMoT i nnotqxHl't

Hac;ri4orc 1. ,flxu1o npsiua

(ciur).

BiOnoai1t: B) 18 cu.

3.49, Ocxiarxu AC $ 3.3. l-lapanelbHicrb

(P-RD). - ;rinis rreperrruy rrroqrnn d i .s@ KVTaMU t=H; Pn=RS ISP =7 -35.42\cn)t PT PR

ll SO (S@

pr =sQ.pn =3'42 =t8 s,R 7

intra cx,

3.48. Pll ll a, orxe,

i npnmifi ix

uePe-

: 10 =3 : 5; CO=+j=6

Bidnoeidt:

[) l6

Ba ABoMa KyraMv.

(cru). CD = CO

+ OD=6 + 10= 16 (cu).

cm.

3,50. Ocri.nrrlr AAtll CCriAAtll BBr, ro sa osnar{oro napa,re.nruocri npffMr4x BB1 I CC1. 3a ymoeorc BBt=CCr Horrpuxyrrr.rx BBPTC iraac 4si npornrerrcHi cropouu, pieni

napa.ne,rr,ni: L{e oanauae, r4o BBTCTC

3.51 . YorzpuH yruux

AATC

napaJreJrorpau

, y srcoro nporn,rexni

i BtCl

cropoHr.r AtC 1

= BC.

i AC pieHi i

.ne.nrni, e napaJrer'rorpaMow, rogi CC, ll AAr. AHa.noriuuo, BB1 1l AAr. ToAi aa ocnanoro naparerraocri npnMux

napa-

BBr ll CCr

3.52. Ilporra:rerxHi cropouz napare- 3.53. Hexaii .43CD pon6, y ;rnoro BD = rorpauis naparelrxi i pinui, romy =AB =AD, MN - tiuis treperuHy a;rorquCD = AB - CrDr, CD I AB ) DPr uu pon6a rrour,r4Holo. napa.rre.nrHoro giaro3a oananoro napa"tensnocri npfiMlrx wati BD. 3a ymororo M i N cepeAunr4 CD C,D,. ctopin BC i DC, MN = 6 cm. Bzcnosor: CCtDp - napaJreJroToti '' MN =)"o, BD=AB=6.2=t2(cv). rpaM.

P*"o = 4AB

12 . 4 = 48 (cm).

Bi1noai)a: 48 cut,

Besriq.

3.41. B). 3.42. A) oAHy.

g.43. BPt npflMa reperr4Hy nroqt4H (ABC) i a. BC 'foai IABC N aA-B,C, 3a ,qBoMa lcyraMlr. 3a yuonorc

ABr: BB, = 3 : 1, ABr= 3r, BBr-

AB. B.C. 4x -^..72 '" '* 1:1

A-8, =

3.r

= 16 (cY).

ia,

cx.'

AB =

roMy BC

"ll

Bpr'

4x' AB BC ""r- "r"rt

cepeAgfl

ninia rpaneqii ABCD, rofi AD ll MN;

BC: MN ltlz*gl=13 MN =AD-* = 22

(cu).

Bi}noei1u: B) 13 cu. CD, 6o ea yMoBoro AD I a' a suawrt', AD reopeMoro @areca N - cepegrna naparrerrbHa rinii neperzHy (ABC) i Q' P*"o = 2AB + AD + BC = 2AB + 2MN' pe MN cepet:ea :rinia rpanerlii. PBID= 2' 8 + 2' 16 = 16 + 32 = 48 (cu)'

3.45. 3a

Bi)noai1a:

48 ctu.

3.46. B) a ll 6.

3.47.8) 3 i

659

t:::i:

3.54. Heiafi P@ riHia neperzHy trJrorqrEu cr i firoil{r4an (MNK). Ba-yuoaoro MN )1 a, rosi MN I PQ. Touy r MNK aa QPK, esiAnn

Bidnoaida: B) 6 cm.

9.44. MN

FEOMETPIS Eincxira O. 9. ra in. Axa.qervrirHufi piBeHb i;ii:1:iiii;ii:i

roNK=(r+8)cn. t-t **t, 1Br=9r | 9x= BiOnoaiOa: L8

3.55- Hexafi s

y! - !!. flxqo KP= r QP PK 72i 4x

72;

cm,

x, lB.

ctut.

trror{uHa, npoBeAeHaqepea xiHeqr eigpiaxa OA

rouxy O. B e OA,

BBl ll AAr Bi4piaorc OA i npani BBri AA1,'rerfiarr, e oAHifi n.norqnHi, flKa reperrrHae uorluHy o ro rpauifi OAr, V IOAA, BB1 I AA' romy

BB'O N rAArO.

04 Buio*r A4- oA- 04 ' BB..OA,_. , ,.(cv); Ll AA, BB,.OA OB OB, BB'

ififiiiiili

rEoMETPlg Einnxixa O. 9. ra ix. Axa4euivHr4i piBeHb

ii:ir

6s9

rqrani, arrfi r{aJreJrcarb i nparri lM i CB (reoperraa 1, uogy.nr 2). A:ie qe HeMornJrI4Bo, 6o sa ymonoro npav.a AM fleperrrnae nJrorquny (ABC), axiit r{a;refirurb nparvra BC.

!-9 ro ("*); =

AA, =

11j

a.(a+b\

4) A4=

B[cnosox: MK i Pq

so ("*),

11. DC=!t*=9=B 2 2

tct").

3.56, Il.roulzna (BMC)

treperr{Eae nrrorqruy (ABC) no 1l BC sn ocl{oBn

12, AB =.5 cvr

rpaneqii.

A,B,

3a osnaroro uapaae.nrnocri npauoi i trJIoIrIrrH[' ocxiarxr.r.4-D ll BC i BC c (BMC), ro,4-D il @Mc).

3.57. Hexafi SABC

- rerpaeAp. S i nepenuuae

rpoxoAr'rrb pelpaAC i BC ocnosu

flror4rana

Ocsi;mnu MN c (MNS)' To Ba oauarcoro napa.nearnocri npfiMoi i naolquuu

nittis IABC

i MN

tAOAr*

oArrosrraarHo Br,r3r{aqalorb

lBoBl

tBoBraa,4BoMa KyraMr{ (zao'd,r--

sH Beprr{r{turbgi,

uuvtpiruni c4nocroponni)' 't"A'

^o e' iiifii:iillil ', a4' oa "d- -4,o -- 5x; oB, = 6xi 5x + 6x = 22; il$?ti:ll:::ll rlosuavnnlo 11x = 22:

x -- 2.

A,O=5 2=lO

(cnr), OBr

[JIolqIIHy. BilpisoH /-B

A e AAr, B e BBr.

#,= H= H ELg 40 u =,2 (cv) i#fiii:::::i'f L\ , a4 = oB =+n = b Fj,:l:::i:::::i:S: BB, OB b t+f:i:ii:iii+ AA Ao I$;li,i:,:*X

-6'2=

lA= lB

,/l /L-A--l / /;' \o B,/

6. E)

2. B.

7. A) ABBfLi B) DCCpr

LE] ls I r

tEtSt

lclr lrlsl L4

[)

AtBtCtDt.

9. B) 10 cu.

BBr tpanv;t

To Boura refi<arrr 6 n o4uifi

2,5

MK i

uorquui.

PQ 6ynn trapaJreJrburiMr a6o nepetuna'nlrcn' Togi rouxra K, Q, M i P 'nexcarr s oAHift nro-

iii:l reOfuerplg SincxiHa O. 9. ra in' Ara4ennirHuf piBeHb :{::::ililiii

cl'r.

cna.

s,o., =9 22l=t.4=12 Bidnoaida: 1

(cru,).

cu'.

AC; - cepegna :rinist IABC. llO = 1 2 AB = 2 . MB = 2..5 = 10 (cu). 3a reopeMoro lli@aropa,4-B' = BC2 -t ACz; 3a reopemoro (Da.neca MO

BC=,[AB'-

AC2

Jt0-6" =fi00-a6 =JA=8 (mr)..ro=;.8=4

(crvr).

Bidnoai}a: B) 4 cu.

- aAQD {BC II QD,6o BC ]I a, a QD - Cx i @D C (ABC). B9, qp AQ'ac (1 F 2)'5 = = = lb (cnn). AQQDABl BiAnosiAb: I) 15 cm.

15. IABC AB

16. AB=lPo"",=4 cm. MN =I^(AB+zK); ZK--2MN -AB=2.7 -4-r0(cu). 4 *'," 2' A) 10 cu. 9ac'ruxa 2

AB.DD. (2+7t.7 7.9 3I.5 ' -- BD22

Bidnoeida: 31,5

Bp _ p4.

AB AC'

(cul.

em.

* .4-B, romy ^ MCN oaCB. k =!.5 * aABC npsuonyruuft, lC = 9O', AB = 13

18. MN

lrlzl

CC,.

10. A) Mn*ro5irtni, flrc6u

660

DDTCTC: B'l

B.[ A

A-ftl If----#

5. B),4A,;

aABC).

ainin, ,4r8, = ! as

cepeAHn

Bidnoeida:

't. A.

4. A) AA.; B) BB,; I)

(erunercrrufi

17. Ocxiamrr DDLllAC,to tBDD.a tBAC. 9acruna

lelt

12(cu)'

TecrAnc caMoKoHTPotK)

(cm). tcu); AB=DC: AD=BC=l-xrw=8=4-',-"','

19. AtBt,BtCt,CrD,AiDr-cepe,qnininiiei4noni4uorpzrcyrnnxinAPB,BPC,CPD,APD. Cropour.r r{oruprrnyrnr,rfta ArBrCrD, bAsiqi naeumi sa cropo:r':a ABCD.

ll .A'8.

AB ll(MNs). 3.58. Ilapa;re.nrni npami AA, i BBl

Bidnoaila:

qepe3 BeprnfiHy ri4noai4no y ro.ncax M i N' M - cepeguna AC' N cepeAtrHa BC. ToAi 3a osrraqerrH,fiM MN - cepeAss

qiii nnorqlrni,

rrurro6in*ri.

Pou"o = (4 + 3) . 2 = 14 (ciu). Bi}noai1u: 74 cw.

nparaii BC. To6ro BC e (BMC)' AD

HaJrelnrrrE,

cut,

AC = 5 cln. Ocxi.nsxrz xarer, ro 1A = 30". To4i

trIo Jrelr{urb trporr{ Kyra 30", 4opinHroe noaorlrni rinorenysu, I

BC=;AB =6,5 cu. MN llAB,romy d MCNa ICAB, ix croponn i nepnmerpra nponopqifiui. Ba yuororo

CM : MA = 1 : 4, ro.4i CM : CA= P^"**: P^"*= 1,5.

p..,,, =Po,* -^cntN 2

-5+13+6'5=4,9 5

cv.

Bidnoai1o:4,9 c*r.

:,iiilitiiiii

TEOMETPI9 SincHiHa O. 9. ra in. Axa.qerui,rnrafi pieexr

i;jlj:

6GI

,4.B retr{arb e o4uifi trJrolqrrHr' flfta reperrrrae rrro ArBr. rrlrany o no nPauift .niuia'rpaneqii A.ABBT Hexait AA, = rc cM' roAi CC1-- Qc + 4) cM' CC,

- ""o"o;" 2CC, = q+ BB,: ,", = 4:out: 2' 2(x + 4)= r i 10; 2x * I --r * 10; x = 2' AAt= 2 cr't' paa:rrit' siApiaox, Ae a, B e a. B € AI<, BM a' 20. AK BM = 8 cm. KM n cr = Q. AQ - riuia neperr{Hy trJrotqEH cI i (AKq. BM I a, touy BM ll A@ (inanure BM nepertuanac; 6n s AQ i nloutunoro cr). * ^BKM ^Af(Qi BK _BM. Ao-AK BM,{4+7) 8 --Lt.2=22 (cwr). AKASBK4 Bidnoaila: 22 21

= (PQR)

a P,.RQ,:

iyt4j-r:.:i......!. idt{f.tvtt.:ft a

!rj:'/.r..ii:.:.:L.i

)ritt:l::tll::.:t: Jl /rli+-..rr.a.:..'. r:JiI-'.r-r-+...r.r...ir \:+if at-rr....{....'. 'idl4JJ:';'.1!a'i L rl al:++.jJ.!..'r iJi4l..-'r::....:.ri .i'lr:r-.:.-......r... L

A,8,, A, € Ocrci,ront

* OB,=?, 5' x+9 =?:,bx=2(x+ 5'-'OB-40

PrR= 7x, PR=

4Qt =

roro lrt ri.nrnr'r ogna. Iloeuaunrrno ir B. Poarrsseuo ga.ni lray n.norqzni B. QO AA, ll BBr; Q cepe4usa AB.

JrroHoK

30cm

P,lft:Pa

9x PR =!!,n

trapaJreJlbsa cropoui-4-B rpzxynnuxa'A-BC, nepernnae AC, Bt BC. B,C = 10 cm.

ftgro no npsuifi

MMr= MK - MtK

8(cM)'

aOBBrea ABoMa KyraMlI (ZAOAT-- IBOB, an lB.fiIc BHyrpirrrni pienoctopouni npr

neprr.rca.nrHi, 1A =

AAr ll BB, i civniit.4S). 4 _ AO _'4,O =?.

BB, OB OB, x

3 cM, ro,qi OA = (.r

- 3) cM'

piBeHb :i:i:i:i:ii:i:i:i;:: i::i: ;EOMETPIS Sinsnina O. 9- ra ix. ArageuiqHufi

(cm)

2 (cu).

2 = 6 (cru).

rrpecreuH,fi n

nlorqnni rpuxyrnuxa-A,BC.

tinir. neperrHy rrolqr4urr (ABC) s nJlotqr4uoro,

[apaJreJrr,Horo npanrift

25. flapa,renrni npaui AAti BBr oAHoBHarrHo Bu3Ttaqarorb lJrorqr{Hy, sxa trepertrHae nJrorqrrHy o no npeuifi ArBr,

floaHaquMo BO =

=I"",.

tlacruxa 3

29, Poar.nsueuo

662

APr - ;rinig fieperr4uy rrrrorrlrurr rpurcyruuxa ABC n.nonlunr a, o -43. AP\ | AB. \ABC - tArBrCr: 48, _ \c = B,C . a_8. _ AB A.C = b" o+c AC AB AC BC 24 10 O6ctrcrnMo: 4Bt =^*lO =8 t"*).

ArB'

(cm).

A,B,i AB, ro rA,CB, ^ tACB'

{f -t; +n,=+-

oOAir;

aisis;

asa;r'oriqse 4o Nb 27.

M,K=\cc,=1.4 '22

24,

cepeass

rq risis: o.c , = !2 da.. - cepear 11 QQ, =QC-Qp=;-(BB,- Az+tl--;(11-7t =2

28. Iloscsesns

= 21 (cm).

Bidnoaida: 8 cw.

aBAB,QC

To,4i

BC - AB 48, toot AB 'Bp aoo Ec= AC,' 4q 3a ynonoro AB : BC = 4: 5,

o'*",

2x+78;Bx=t8;x=6.

27. {epea napare;mui npaui,4A, i BB, npoxo4zrr, naorqrHa i go

MK =! DD, \.16 = 8 2'2

9); bx=

OAr= 6 cmi OBr= 6 + 9 = 15 (cM). ArBr= OA, + OBt=6 + 15 = 21 (cu).

V aAArB, Q,C

$,;r*........ 23. flaorquHa, uii,'r-$::i$:::.* {/,:/rJ...a\.niJ. i$i.J,rr.-.-.....-.-..

.qzs. .4o J$ 25.

(r + 9) cu.

ct).

22, [us. puc. Ao aaaa'li 21.

2t,

OAt = x cu, OB, =

^ Ptq. PR .Q.n _ Tn= en pg

Hexaft PP, = 2x, PR = 3x, ro4i PrR = x' P.R PO x.30 p.O.=:t::--:1 =';" = l0 (cm).

PPr=

26. Puc. i noscuesss

ct:tt'

. PQ ll P,Q, (P181 a PRQ

*-3 3(r-3) =2x;3x 9=2x: x=9. r =?r 3' AO = 9 cu; BO = I + 3 = 12 (cu). AB = 9 + 12 = 21 (cv).

9. flparvri,4A 1, BB\, CCli niapisolt

AC (u).

K e AB, M e BC, KM = 4 e\r, AC = I cv, MC = 15 cu.

c (ABC), AC 11 u, rorry /C I KM (sa o3Haxoo trapa.nearnocri npslroi i n.noqzuu). r KBM.a \ABC (aa gnoua xyrauz),

KM BM rro3'aqrBr tnBM=r.MaeMo: != " g r+lb' AC= BC; 4(x + 5) = 9x; 4x + 60 = 9r; 5i = 6O; x = 12. BM = L2

cm: BC =

BM + MC = L2 + Lb = 27 (cv\.

ganuit rprrr(yrxrn; M, N, K cropin.4B, BC, AC nit30. Hexafi -43C - cepeAunr4 noni4no,'rlepee rouru A, B, C, M, N, K npoBeAeMo napa.ne.nrni npanti AA., BBr, cc\, MMf NNf KK|

NN,

rpaneqii CCPTB.

- cepe4n,s .niHis ll =? NN,=CSEB'i cc,=27tJ1,,1 ,-BB,=2.g '2 rpaneqiiACCA. KK, cepegua.riHia - AA+ AA. KK, .'--t,---:-:t: M, = 2KKt - CCt- 2. 1O 7 = 13 (cu). MM, cepeAns ,rinis rpaneqii MB\r. * Bq 13+ l1 twttt. = 4 =24 =12 kvt\. '222

:ilt{t:iit{

TEOMETPI9 Eincxixa O. 9. ra ix. Axageuivxnn pieenr

illi;

663

4,1 2.

poM6, a I BD,A e a. flpaua a i npawa BD oAHoBHar{Ho Br,rrtarrarorr, flJrolqnuy. HagseMo ii o. 4 e tt' O e rx (ocHinrr*nO e BD), ro4iAC c cr,6o 4ri ro'rcuA i 0 npnrvoi AC lexa'ru Y nloruuui o. Ceo,,DE0,roMyCDcu. Oexi.nrrrz npnlri a i CD .nexarr, y nlou4Izui cx. To Borlu

91. ABCD

4.14. A) (oana). ,{.nn no6y4oeu qiei rounu (4auo urluo6ixr+i npaui

aib):

1) qepeo 6y1t'rrr+y ro.rxy qpamoi {z rpoBeAeMo npfi-

ne e mnmo6ixnalru. flprznycrzmo, ryo a i CD napalenrxi. To4i n nlorquHi c uepes rovrcy D trpoxoAurb ani pieHi np.aui BD i CD, napalelrui np.alrifi a' aile \e HeMoltcJrr{Bo' Orxe, npaui a i CD nePervuarorr'cfi.

32.

Ba yrvlosoro

a Ji BD, rouy a i BD nexarr n o4uift

uluni. Hasseuo ii

vy b,, b,

) b; 2) vepea npami a

n:ro-

cr.

u J,ir';f..-...4..j... rJrn...r-.4......t!

uJrT{.:'r...4.1.!t. trd^ jlaat-.-i,....'rr, \- Ji[.it^.-.....f tr.. L.r6r'11a..+;.f...L i.n^}t ....'...i.'.'t.

4.3.f)

.8) flapa.nemni.

uoro

II.

1, 3, 5, 8.

B 2

T 1

T 2

4,1 1. B) flapa.ne,rrui.

4.8.8. r, lI. aa reope-

Moro 1 (MoAyfib 4),

664

il:i: ;EOMETPIS Einsxina O. 9. ra ix. AxaAervriqHri piBeHb

napa,re.mni,

i a cc{. IlplrrrycrrMo. rqo fipflMa d treperrzM cninrsa rovxa n.xorquu o i B, -

npnrvri,

nlo fleperr,rnarorrcx, aa arcio-

saAarcTb rrJroil{rrny.

4.18, Hexafi O

4aua rouxa.

Poara,flHe&ro napr{

ocl\i

rpr{r(yrHnxia OA.B, i AAB, OBpr

ocA.

nogi6ui. 3 no4i6Nocri Br{[JrrrBae:

4.10. A.

KZ ll AB, ZM ll BC,nx cepe4ni niriii a-43D i yCBD. (KZM) ll ('4BC)

Ocxilsrcu nloulr,tun napa,relsHi, ro qi rpaxyrurrcr,r

4.9.B. B

trJrorqr,rur4 cr

3a osHaxoro napa.ne.nrnocri fl,'ronlux (ABC) il (A.B.Cr).

i oBc, A

npoBe,4eMo .repes

fonegeno, qo (ABC) ll (A'B'C'). Ocni:nrcu O cepe4r.rua rc6xnoro e ni4piamin AA' BBr CCr ro-IAOB = aArOBr Ba fiepmoro oaHaftoro (IAOB = ZAtOBt nn reprunanrni). B pienocri rpanyrurzxia BrrrrJrrrBae: ZOAB = lOArBr ro4i ea oanaxorc ilapaJreJrbuocri npaunx aa nnor4uui AB ll ArBr, Arra.noriqno ,4oBoArrMo, ao AC li ArC'

Ogsana rapalelruocri rrJron{ilH Tr. flrn1o 4ni npaui, ari treperuIralorbcfi, o4niei nloqnnr'r, ni4noei4no napa're'nr'ni 4uolr npanru* 4pyroi nloulrnr, ro qi nJrorqrrur4 lapaJre,'rbur'

4.7.

urruo6incsi, BN6epeuo Ha npfi-

Haâ&#x201A;Ź rrJrorr{r4rry p e 4eaxifi To.{qi M, ro4i rouxa rqo cyreperrrrrr, ymoni cr ii B. Bucnoeon: rrpflMa a He rreper[Hae u.riorqzny B.

4,17. Itonrsi gni

5.A (oauaxa napa.ne.nsnocti n,roltrun).

4.6.A.

n.ro-

1l c. 9epee nparri a, i b npore4ervo rrJrorrlriny o. .(ri nparrni a i br, tqo flepernuarorbcs, urorrlr{Hu B ni4noni4no napalersHi AEoM trpEMuM a, i b, rqo neperr4Harorbcs, rrJrorrlr{Hr.r o, 3a oaHarcorc napa.ne.nrnocri uJrotrtr4u cr ]] 9.

4,16. Hexafi

4.4.A, B, L.

4.2.A,8, [,.

npruoi i

uer upauy br, napa.ne.nruy b, (rlepea rortcy, slla rre Jrexc[TE xa 4airifi npamifi, Morr(Ea trpoBecru e4r4ry rp,fiMy, napa.uelrriy 4aHifi.) Ilpami c i b, aagdrorr eAr{Hy nJroquHy B, 3a oairarcoro napaJ'reJrbHocri npauoi i nron{znra b li B. Ana.noriqso [poBeAeMo npfiMy dr vepea goeinray rou-

npocropi' flapanenuni

Il.norqranz y npoctoPi Mo]ltyrb: a) neper:zHatraca no npa*tiii, srcfiIo BoHtr Marorb o,4ay cninrny rovrey; 6) a6irarrca, .fircilIo Y Hrrx e: npalri, u1o fleperrlHalorbcs: - 4ui cnirsni cninrna nprua i Torrxa., rr]o iii ne Ha:!exurr,; - rpu cninr,ni TorIKu, rqo He rblfiarr, ua o4nifi npgrvriii; -e) Syrt napa.ne Jrbluvrtt, To6To He wratn rxo4noi cni.nsHoi ro'rxra'

it'rni,..+.....r."..v.

aa osnaxoro rapaJreJrbHocri

xy C: a,

nroull4Hll *rjnf,.......,.....,1

b ll c

(x.

uifi a (axc, I.r) Aoni,'rrHy ro.rxy A i

Mo.q,vnb 4. B3AeMHE PO3MltrtEHHg nnoll4ylH v nPocToPl

liir!.,........ i4?i.....'.:....1.

flJrou{ilHy

4.15" Hexafi nprui a i

Bsae&lHe posMiu{eHHc ABox nflou4l4H y

ibr, axi fleperrHarorbcfi, npoBeAeMo

e/4,rrrry

qn{u.

A e o, rovxa neperuny 4iaroua-neft ponda O,rexr{Tr' y nJrouluni c, 6o narerxzrt nPauiit BD. \s ,{ni ro.rxx 4 i O npnuoi AC sa,rerxarl nrorqzxi o' roMy AC c a (reopenila 2, rvroxytn' 2). Orrxe, pom6 ABCD i flprrMa 4 Jre]+(arr, n o4nifi n.noqrzni, a nparvra b - s isuriii. IIe rtralro6irxni. o3uaqae, rqo npxmi a i b

94.1.

A) fleperauarorrca.

4.13. B) S6irarorlcs (cni.nsni npama i roqxa).

:i:::i:i:i;i:ili:i:i

04 Bp1 _OBt _OCt. OB=44 AB =O4. OA, BC OB OC, o_tq AC=oAa AC OA=4c,. 3 rlHx nponopqifi uacuo, 'ABACBC 43 = 49 = ry. ^o (nponopqifinicrr, I{e oauavae, rrio sa rperr,oro oanaxoro ni4uoni4nr,rx cropiir) a-4IlC N aArBrCr.

iilti;litiiii

TEOMETPI9 Einnxira O. 9. ra ix. Axaseuivxui pieenr

ll:!

665

9. gepes Beprnr{rrr4 trapaJreJrorpaMa r -BCD nponelr'r npami.4-A, 11 BB, ll Cc1 ll DDr ari neperny.nu n.norrlr,rHy c B Toqrcax Ar, Br, Cr, Di. roqria neperrHy 4iaroualeii, -BCD' O Hexafi O cepe 1nlr,a B D. Ilpoeegeuo OO I ll B B I ro4i sa reopervroro cepe4una BrDr. Ana"norivHo MotrcHa Ao@a.neca O, trapaJrecepe4uua AlCr A\BrClDt Becrlr, rlo O, cepeAHs .niHia rpaneqii AATC.C JrorpaM. To4i OO,

3a yuoaoro AAr = 4 m, BBt = 3 AA3 rpaneqii AATC,C: OO' =

3 BBTDTD: OO,

u' CCr = 1 rr. +CC. 4+l =; = 2.5

T BB tDDt. nn Ln , --t -t.,n =

Hux pieui ri4pisrcu,

3. ,{ni n.noqunn, napa.ne.nr,ni rperifi, rapa.nelr,ni rri*t co6oro.

4. flrorqnna, .rrra fleperuHae q.4uy o ABox naparerbHux n.rroulrtH, neperunae i 4pyry. 5. Ilpaua, .flHa rreperlrHae oAHy B ABox napaJreJrrHrrx fiJrorqr]H, neperuHae i lpyry.

4.23.8. (m\.

, r2 - . --t -r.

2,5-3=2(vt).

rrJrorrf'Ila. i.r*u t o( - AaHa - 4ana Ilpone4euo qepes roqxy B n.norquuy B, napane.rrny ulorqrani c.

.2o.HexafrB Hexai

p, qo

rpe6a 6y.ro Aogecru.

$ ne MoJ+{yrb neperuHarr,rcff. .flrc6r ninorqznu u i B rra;rz cni.nr,iry Toqxy, To qepe3 qxr roqrcy trpoxoAn rn 6 Asi nJou-u{Hfl (o i 0), napale,trti nlouluui y. A qe cynelpeuzrr reopeni 2 (uogy.nr 4).

4.22. l\ Ocxi;rrrrl M i N

4.g2.

AC =

B) AC = 15 cu.

4.30. A, B,

.U.

4.31.

AA, i BBr, ro MN I AB. flpz no6ygoni nepepisv ny6a n.norrttoro (MON) BpaxyeMo, rqo O - rovxa neperuny giarona.nefi rpani DD i (ABC) rapa.nearHi.

,C ,C .

axi,

BC'z =

Jrs +r44

t69

(crvr).

4,33, flapa.ne.irrni n,rorquuu o i p neperznarorbcfi B napa- reJ'rr,EtrMrr npa:lrutttuAA, i BB, i ni4runarorr nig trr.rx pieni nigpiana: AA, = BBr. Topi AATBTB - iraparrerlorpaM (rpouurextni cropoHrr piani i napa.re.nlHi) -4r8, = AB = a (an nporurelcHi cropouu napa.ne;rorpama).

4,34."|at<. Hanpux;rag, fiHrrlo oArra B napaJreJrbHux rrJroulrrH lricrrrr ocHoBy, a Apyta 4,35. (FPQ) ll (DrBrcl 3a o3naxom napa- 4.36. floscreuHg aua"noriqse ,re,rssocri uroil{r4rr: FP ll DrB, ax ce,qo N! 4.35.

lltrot4unlz (M NO)

Il;rorqrzna (BDM) npl,z' nepetuni ry6a yrnoproe nepepia s rpauefr AArDrD, AATB.B - tMDB. Ito:xna i ABCD Mae B rroilIrauon (MDB),4ni cni.nrui rovHrz'

JAB\

-rreperuHae cropontr pinuo6e.Apeuoro rpr,rfiyrnlrna.

ea yMoBoro e cepeAuHaMrr pe-

6ep

B'e4uanmu

A, B, B, [. I A\ fleperuHarcrrca. A, r.

P*,4", = P*" = 5 + L2 + 15 = 30 (cu). Bi0noei}u: B) 30 cu.

4,2i,fIloqnnr.r o i

l:iliti:!::::i:::::: ... ..... .tl\rr...

4.29.

A, t.

4.24. 4.25. 4.26. 4,27, 4.28.

,4oni.nrua npsMa, rqo npoxoAurb rrepe3 Toq-

xy B napa.nearso ct. Bisrueuo n n.norquni cl 4onillny touryA i npoeeAeMo qepe3 tounyA i npsuy b rrJrolquny y. To4i nJrorrllna Y fieperunae rrrorrl[Hlt a i B no rrapaJreJrbH[x np,fiMr4x aribr, ale np.alra b, trpoxoAr4Tb uepes rouny B, a npsMa b raxorx .nerxuru y nnorquui y i npoxo4ntr vepee ro'rxy B i sa xac.:rigrom 3 TeopeMr'I 4 napa-ne.nrHa ripalrift orTogi aa arcciomoro napa.nelr,noeri npaui b i b1 Marorb cniBua,4arrz, rouy b $tnl,'-'............

nnou4r4H

1.,flnqo 4ni napanenrni n.norqunra neperrryru rperr,oo, ro npaui Ix neperuny 6y4yrs napalelsni . 2. flapa.ne;rrni rrJrorrlt4nrr, neperzHaroqu gni napa-neltni npaui, ri4rlrirarorr ni4

i BBlDtD.

4.2. BnacuBocri napanenbH[x

pe4ua rinia a DrBrCr, PQ ll BrC art cepe4na ,rinia aCCrBr.

orpuryraervro nepepi3.

Il.noqranu (MDB)

i (Bpp)

fleperuHarorr'cs.

2) flo6yaycrvro ix ,riHito fleperilHy. MB n ArB, = Xi MD AtDt = Y. ^ XY uryxaHa ainir nePerunY. uryxaunti nepepio. 3\ DDPTB -

AAr ll BBr roMy Ba osHaroro.AA, Ll (BB'D,). Sr4o,, = BBr' BrDt' BrD, = AtBr' J2 @nucrunicrs giarona-neii KBaApara: a = ali ).

Bp,=3^,15.'E=6 S"u.,,, = 3,1r.6 =

4.37. Ocni.nrrcu IMAB + IAQP = 180", a qi nyrz piurHi o4nocroponni nprE rpflMux AB

roAB

(cru).

tavE (cv').

:::# rEOMETPIS EincHixa O. 9.

ra ir. Axa.qerurivHrfi piBeHb

I PQ.

(ABC) il (QPK) sa osHafioro napa.ne.nsnocri

i:i;:::::J:lia:i

e BHyr-

ci.ruifi,4@,

IMKQ = ZMCA, a qi xyrz rpu - Iri4uoniAni AC i QK i ciqsifi KC, rowty AC QK.

666

flpfiMr4x

rrrorqrrn.

i:iiiililiitlii TEOMETPI9 EincxiHa O. 9. ra ix.

Axa4ervrivxrafr pieeau ;|:!:,i 667

4,42, flpami OA2 i OBz, rr{o neperlrHarorbcfi,

i Ff ll BD sx cepe4ni .ninil ni,qnoni4xo yADC i aADB. Toai (TS.F) ll (BCD) 3a osganoro napa.nelr"nocti nno-

4.38. FS ll CD

rorb n,'rotqlrHy, sxa neperr,Inae rrJrorr{r{Hrl o paJreJrbHr,rx flps Milx A\Br i A2B 2. Ocxfursrcu

trI11H.

BrraHarra-

B no na-

A)Bt ll A282, ro tArOBrN rA"OB"

, 40 _ 4,8,. ' 40- A"B,'

(r4-n -

aB. =16_i9= 2o

48,. 50 '

(crur.

/,,*A",f

4.39, DPl ll DB sx.nisii nepernuy napurerrr,Hrrx rpasefi trJrorrlr4lron DDtBrB, floaga.rfiMo pe6po ry6a a. To,4i n aAr-LDr i tKBC ArD, = BC = ai

o',L),",

4.43. a)

a-L-- KC =!a.

a KBC 3a ABoMa KareraMrr' 3 pieuocti rpr'rxyrnrlrcin LDr= /f3. Ana.norivno D, K = BL' IJ,e osnavae, go LDrKB - ta-

tArLDr=

paJreJrorpaM (nporr,r.irexni croponrr [onap]ro napareJrb-

ni) i D,K Ll LB. Bucnosor: (Bp$) ll (BDL) 3a o3Hatcon napa.ne.nrnocri rrlorql4H (,qni npsui, rqo

DK - DrBri

neperrruarorl,cs, oAniei nJroilIrlrru ro1

napa.tenrui ABoM [psMr4M Apy-

-DBiBL'1.

I, MK I NL. Togi, epaxonvroutr, rrlo sa BJracruBictro napa.ne:rnrrx nJroulrarr MK = NL,

4.4O. Bi,4oryro, qo cr ll

MNLK MK = NLl.

po6rauo Br{cHoaon, n\o

l#l&;ri;rfin XtfriiJ.:::i:iJ:: [5]f$l:j.;:;:;:;;j

ffififfii,*jj't

(MK

NL.

Hexafr.

ML = 14

cw,

NK = I

ct'r,

- i, tBPC. pe4ui ;rinii aAPB Korxna a,nap npauux AP i PC ra KM i I{N, rqo neperlruarorbcfi, BrraHarrarcrb Aesr{y n,aorqr,rny. KM ll PA, KN I PC, rony (MI{N) ) (APC\.

rapaJreJrorpaM

MK : MN = I : 7'

4,44.Poss'aaanrss asa,rori.rne 4o Nr 4.17.

rouxa neperuny giarona.nefi napa"re;rorpalra. ToAi

MO=OL=l ML-7 2

4,45. BrarouaeMo AoBeAeHH.rr, uo6y4ynannrr.r rani nlorqnnn. PoarrsneMo rpz nonapuo ur.ruo6ixui pe6pa ry6a

"*,

NO=OK=1NK =4

cm.

Iloasaqruo NM = 7x, MK = 9x, INOM 3a reopenroro xocr4nyciB: MNz = NO2 + MO2 - 2NO '

MO 'cos

s';

(

(7x)2-42 +72 *2.4.7 coso") .l (1) 49x2 = 16 + 49 - 56 cos a. t MK2 = MO2 + OK2 - 2MO 'OK' cos(180" - a); rt (2) (9x)2 =72 + 42 + 2 7' 4 cos s. (r > 0)' ,{o4auo (1) i (2): 4gtc2 + 8lx2= (16 + 49)' 2; l50x2 = 130; * = 1

MN=TcwiMK=9cw. .P*r"*= (7 + 9). 2=32(cnr). 4.41 . Ilpmri OF i OP, qo fleperr4narorbcfl' Bllsrraqaror], nJrorqr,ruy, .fiKa neperuaae flJrolqr'rur{ c i B no napaJleJIbH[x np,rurax FrP, i F rP r.'I opi o F rO P, t t F P r, "O OF. F.P OP

seiwu

9= t :B(x+4\=sx; 5 x+4' 3.tc+L2=5x;2x=12;x=6. OPr = 6 .*.

668 iii: feOugfplg

[ni npsui a i b ne e

6 ArA"

napaJre;r;Ht4Mr.r,

r:oAi

= BrBr, a ca yMoBolo noirrz piaHi.

.flrrqo npnrvri a i b neperunarorrc.fl, To rrepec Hr{x Mo}r{Ea flpoBecrrl urorqr4uy,

sra fieperr{Hae ir.rorqunra ArBr A2B2i ABB,.

cx, B

y no

/A.

rrapaJreJrbHr,rx flpffMrrx

3a reopelroro @a,reca:

Sincxixa O. 9. ra ix. AragerurivHhfi piBeHb

ABCDAIBLC.D1 AlBr DtD i BQ. flo6ygyel,ro repepiau xy6a n,rorqlrnavu (BrDrC) i (A,BD). I-[i n,nouluux Maror.t 3 ai4noai4nrur.r rpanflMu rcy6a no 4ui cni.nrni Toqlifi, a orrfie, fleperr,rHarors qi rpaui no flp.rrMr4x, rqo flpoxoAfiTr, vepee qi To.rxfi. TaxuM trrauoM, ABTD'C i a BDA, - ruyxani nepepiez, BD ) BrDr CDr I BAr ax 4iarona.ni nporra.nerxnrax rpanefi ry6a. To4i sa ooHar{ox) rapa;re.nrnocri nrrorrlrH (Bp1C) ll (BDA). L{i napa.ne;nui fiJrorqlrHlr rigrnnarorr sa mr.rlro6irnnrx npsMux pinni si4pisxu ArB' BC i DDf

4.46.

oi= ,,r,= *,

Hexafi AP, = ,c, ro,4i OPr=

3'e4naemo rouny K g ro.rnauu .F' i M, a ix is roq$orc N, .qe.F, M i N sianoei4no cepe,qrrrru BiApisniB AP, CP i DP. To,qi -FK ll AB, MK I BC sx cepe4ui- ninii rpurcyrnuxin APB i BPC.3a oananoro napa,'renrHocri nJrorrltrH (FKM) ) (ABC). 6) Hexait M cepesuna AB, N * cepeAuHa BC. 'foti MK'11 AP, NK il PC sH ce-

3a vMoBoro: ' BrB" =

5.6

:i:::li:i:::::li:::

:ilil{liltil

B.B^

BrB'^

4A,

Arn,

ArA"= BrB" 62 B,B" 5'

lt) (clt); B,B" = 6 +

I"5 = 21 (cu).

/to,l-

L)",i

?:,/

1f)u1,""1

TEOMETPI9 Einqxixa O. 9. ra ix. Axasewrirnnft pieenr

fi!:

669

4.61. Hi.

5 2 5 44 =!-5 5' 4L, =2-; A"A.=?' A'A'

= rz.s (cu): Ar4,

12,5

= 17'5 (cru)'

npami a i b nurro6ixni' ro qepe3 rouxy Ar npoBeAeMo npauy bt ll b' sxa neperue t"rio.,tr""t B i y o rovrcax C, i Cr. Arb"= BrBr, crC"= B"B" an riApialtrz [apaJreJlr,Hux fipflMr{x' o6mexeni rapaJreJrb-

fxrqo lx HI{M

h nIorqIIHaMl4.

,{ari posr'aeanufl Bi}noai}a: 17,5

sBoAl4Tr'cfl ,4o Blzna'4ny, Hoau

crvr

e 21

npaui o i b ueperr.rnanrr'ce'

cN{.

nnoqnni pficynxa eiApis-

2. flapare.nrHi eiApisxz @irypt eo6paxarorr'cfl rra nlorquni pr4cyHxa napaJleJrbei4pieramz.

3, Bianon]eung ,4oBlnr4u ni4piaxin ogniei npamoi a6o napale'nrnux npaur'rx a6epiramTbcg.

t, L. f.

4.53.

B 2

4.64. L) flpoenrlis a DB.C na rpanr

6yAe

B,CTCB

2) flpoexqia r DB.C ua rpans ABCD

lrc 22

=\.6.

ei4piaox -BCD.

BrC. Ar

18 (cur,).

ArBrCrD, 6=l8 (cu'z).

r BrCrDr.

piauocroponniit rpratyrnzH, rArBrC, 4.65, ABC - fioro npoexqia. Y aABC BD LAC, D cepe4rzna AC, M cepegutaAD, KM LAC, KM .

4.54.1. 4.55.

Ocxillxu npft [apaJ'reJrr,HoMy fipoex-

siAHome

B,C, " fl.

"*y

lff,H:Xll""ru

nPoe*Ilirc

tuct""y crasoszrr AC niqAB (AtC, ni4 '4tBt)' 4.58.,{ur. puc. Ao aagavi N! 4.57. flxttlo

(App) |

(ABC), ro AA, = BB, = CC,, romy A-ACCr, CICBB. ATABBT'- napa' ,TeJrorpaMlr (nporra.nencni croponu pieni napa.ne;rni).

ArB, -- AB, B1C\ = BC, ASt = AC ' tArBtCr Ba rpboMa croPoHaMI4' = 'ABC

SeiAcra

nPnmoi

BD.

/l\ ^

\ C

8,D..

y potn6i ABCD AF uryHana eucora rve4iaua pirHocropouHroro tBCD.

4.66.

l) nponecrr ve4iagy BrD, rA,BrCr: / I | / h L 2)poa4irurnaigpiaoxAr.B,raurir' A M D rouxaMr; 3) nponecru M,K,

uimeaHHi a6epiracrsca nponopuilinicrr, ra napaae.nr,nicrr eilpialcis, AC

Ha rparrrzqi epyvno sHafirz,

4.57.

1) ArBrCrD,; 2) Dpi 3) AtD; 4) A,B,C,D,; 5) AptC\D; 6) Api 7) r BrDrCr; 8) a-4,-B,D,.

tt S*,".o =tBtct qDt=t6

IIIo6 nznonaut qe saBAaHHs' Tpe6a ctroalarny BctauoBr{rz nigHourenna BC

Tarun xe

ce-

4.63.

3) flpoexqia r DB.C ua rpaas I

E B

4.49. D ni,qpiaox. 4.50. A, E. 4.51. A, B, B, A. 4.52. A, B, f, A, E. A

o4HoexaqHo Brauaqarorr, [JrorqlrHy, fiKa fiepeTlrnae nJrorqr,rHy o no rpxuifi -4r8,. Tomy, axrsqo AB a = C, ro C e ArBr. 2) IIpw napaJreJrbHoMy^ npoexqiroeauHi a6epirarortcfl nponopqifi aicrr ei4pianiu o4Hici npauoi,

s^-"^ =

4.48. A, B,

4.56.

4,62. l) Ilpsui,43 i,4rB,

AD ^, A,D. = "' DrB, DB'

KAMI{.

4.47. A,

paJTeJrorpaM.

pe4una ArBr.

i npocropoeux OirYP Ha nnortlt4Hi

arairnu

ni4piarie

rouy eo6palrleHnsM Toqr{u D 6y4e rovna D,

$4.3. l-lapanenbHe npoeKqiloBaHHq' 3o6paxenxn nrocKlrx BJracrr.rsocti eo6pamenua npocroponux (piryp i. flponto.rrlriiisi ei,4pisnu 6i"ypz so6pamarorbcs Ha

Eo aa npaenramH napaieJbHoro npoenuirosanri,r npoexqii napa.lelsuux pisu.e napaJreJrblri pieHi eigpiaxu. Tanzrvr .rznom, ao6pa:nennavr napaJreJrorpaMa y BaraJrr,HoMy BlnaAHy mae 6yrra naHr.rx

(1A = 60"). A.F ll BK,

p,e

nncora

fIo6y4oea:

l) AtBplDr

rapa,'rerrorpaM

eo6paxenna pov6aABCD. 2) K, cepe4rana ei4piaxaCrDr. 3) BrI{1 BK. 4) Af

- eo6paxteuHa ezcoru 1ll BrKr- uryxarre eo6pa-

}

lxellrrs sucorz -Af.

4.59.1.

Ilpoexrliero npaivroi e npsMa' a npoercqiero eiApiaxa

2. Ilpoenuii napaJreJlbHhx rrpflMilx naparelsni' 3. Bignotreuun AoB)rczH ni4piexia rapaJreJrbuu x

rrpfl Mr4x

eiApiaor'

a6epirarorrcs'

4.6O.,(.nn no6y4oera 3o6palxerHfl rerpaeApa aocrarHbo no6y4ynaru 4osinsnrafi tpurouxy' KyrHr{K, rqo ao6pantae ocHoBy rerpaeApa. Iloriu tpe6a nu6parra arcy-ne6y4r rxa6y4eao6paxaruBepIiII4HyTeTpaeApa'is'egnarr.riiauepurr,rrralrl{MIIoIoRyTHuao6pa:rcenHa 6ivnrzx pe6ep' xa. Orpurraxi ni4pisxn

670

i;:::::

rEOMETPIS EincxiHa O. 9. ra in. Axa4ervrivHnf piBeHb :*1i:iiiiiiiii;i

4,67,Hexaft ABC

4aHzfi npauorcyrHufi rpzrcyruux,

lB = 6O", BD -

6icexrpzca

rcyra B. BiAomo, rqo 6icercrpzca Hyra rpr,rT(yrHr,rxa 4i,rrzrr nporrrJrex(Hy cropoHy ira ej4picrcu, npouopqifini npr.r.reuzm cropoHaM rpr*yrHtrKa:

flri rarer, qo

liiiittitl{i

30'. Togi + :'+ "CDBC|

Jrercr4rb fiporr4 Hyra

+CDBC2 = *4! drc cn =! nn =

TEOMETPI9 EincHixa O. 9. ra ix. Axasenivxrfi pieexu

f:id

671

9acrl*ra 2

Orxe, rqo6 no6ygyrarra so6parxenua 6icexrpucu BD, norpi6uo rla 3o6pax(eHHi ftarera

AC nossaqururovrcy D, rarc, ilIo Crnr=f,4o' i o'egiraru ii

\K l'\

s Br.

^l \

"U;P"

.L \"

"l---,-

KO _ RH OP CP

18.

9. E.

2. B.

10. B.

3. B, A.

11. r.

12. E.

5. A.

13. f.

6, .lleperunarcrr'cs a6o napa.ne.nr'Hi.

14, A)

4. !i;il,\'r.,-.'.'..-..'...:

$fi$$*i*i

Ififrdiitriiii ii*fr'.iltirri.ij.,iii r;;\:i'*.;.:.;....1i

8. A.

ca2- cBz' Ocrci.nrxr CA,

T,L)U,'

-!, "Bt 124

Ay', = 1 : 3, ro CAr: CAr=

cB- =

fl.norqr,rna (/3S) neperunae n.rrorqaHrr o i B no npa*rux ArB, i A282. ArBt ) A2B2ll AB. Ba yuonoro A, - cepe4una AS, A1 - cepe4r.rra ArS. ToAi sa reopeMoro (Daneca B, cepe4rana BS, .B1 cepeguna BrS,

ArB" iE(o,qnor.

Inarnre ri.norqr.run

15. B).

- cepeAuff .ninis a-A-BS: t8=9 (cu). 4r-,=+on=1= 22

ArB,

""p"g"*

,BiOnoaiOt:

;/iNll.il-..+r:...r....

niiria aArSBr:

+n,=\+n,=1 2" 2 4,5

s=4,b

(cu).

cttt.

2O. OcNi.nrxu (AlDpt) 1 @DCr, ro ArDrll AB, Dptl DC, Bpt ll BC, Apr I AD. DC, rowy ArDr, DlCr BtQl iArB, D, - ce- cepe4una peAni ninii niguorigno tADS, DCS, BCS, ABS,

16.

=l2'dO= 4 em: B,C, = ArD,= 4 cwi c,o, =!co =lan =44 = s (cru). 22 n O,

oK = OH= HM - roMH: OM 6 K! . - 6'10 A)'HM -oK OM 5=o!10 o* 5 -I2 ooK-L

P+a",o, =

(cM);

KM = OK + OM = L2 + LO = 22 (cvr); oK B)'oM =aL : oK =oJ:gy=t-? =B Gvt; KM=B +

+ 3)' 2 =

(cm).

21. Ocxi;rrxrz a ll 0, ro.4-B

lz(cM)i

ToAi

oH 12 oH 9= Kt B).HM =9K, ox=[!:9M HM =34 6 Gwl;KM=6+8=14(cu): OM y1 9L =9L:6 =J_ ; ou =9-q=12 (cM); KM=i2+8=20(cM): 'oH 6 oM 9 0M' Einsxixa o. 9. ra in. AxaAeMi,{nni pieeus iii:i:iiii:iii

APr

aASB- aArSBr, aBSC- r B1SC1,

APr

BC ll Bpr

^,4SC-^AlSC1.

3 no,qi6Hocri tpnrcyrnlrxin Brarrr'rr.rBae: s4=A,4 =54. Bp1 =54=SC'. sB2- A2B2- SA2' B2C2- SB2- SC2'

s.4,

4c,

sc,= s4= 4c,

KL (cu);KM=6a4=10(cM). n\ HM =oK, --. 10 KL =!!5=6 OM. oM-oK:-gM

672 il:l rrouerplg

3 qax nponopqift maemo, *o

i::ilti{llil

rEo

22 vcirnp.lo u.. q.

M 1

ErPl

: 4.

3 (cru).

19.

rra.tu 6 cninrni Toqxrr, a qe cynepeqzrb yrroni (o I F).

iil

94_94

1. B.

Ilprui A4,2i Bp2(A4r^ BrB"=C)

reperrrHalorb riapa.ne.nriri rrJrorr{r{Ht4 o i B no upaMrrx ArBl i AzB2, ArBr ll ArB2. To4i a,4,CB, N a ArCBr.

Tecr nflq caMoKoHTponK) 1

.. op _ KO 'CP _ 9,6 . 4,5 = 3.2 (cv). KH 13,5

/ ",D:'/ 9acruxa

17. fl.norqrira, ,fi$a fipoxoAr4rb .repes uparri KP i CH, qo neperr4Harorbcfi , neperrrHae napa"re;nxi nJrorqr,rnra o i B no napaJreJrbnr4x np,tMrrx KH i CP. yKOH * aPOC sa ABoMa ByraMt4 (IKOH = IPOC sK Beprqr(aJrbHi, ZKHO = IOCP ax nnyrpilrni pienocroponni npu KH ll CP i ciunift C,FI).

t+. €48,= ++ 4C,= B,C,

s Einnxina O. 9l ra in. Aragerui,rnnfi pieenu

i{!:

673

r A

aHauurr,, Ba rperboro oguartom no4i6nocti r:puxyrHrxin (ea rpr,olta croponanrr'I)

aArBrC, a tArBrCr.

.sA_ 3avmonoro "sa23 f= Maemo:

P*,",., Pno^^^"^

Toni

48,

p.. ^^ =

=?.

4C, _

BrC,

_?.

48,= 4C, B"C, 3 Pon

(8 + 10_+ 12). 2 _ 20 (cnr). "=""'2 _ DO

27. Hexafi AtBrCtDt npoexqia rpaneqil ABCD. Ocni.nr,rcr npfi napaJreJrhHovy npoenqirosasui a6epiracrscs naparersuicrs npflMux ra nponopqiituicrr niApisRie o4Hiei a6o trapaJreJrbnrx trp.rrmux, ro ArBrCrD,

rpaneqia, BrC,

8x=24;x=3.

Bilnoei}u: 2O cw.

22. floscseHsa i puc. gue, go N! 21. flic:rs 4onegeuH,s noAi6nocri rArBrCri tA"BrCrpoBB'flBaHHff Mo)fie 6yrrr raxrrM: 3a ymonoro s,a! : s-a! = nb

=#=!!i s.,"- = A,c:h "M,4', 4 =q'tr 4

,2,

3ai4cu,4,c,

= = 'osi ++ 2 B"C,= + 48, ++ 4C, Y h =6 (cm).

MtOt= 3. 3 = 9 (cm); O,N, = 5 . 3 = 15 Bidnoeidt 9 cv i l5 cu.

r{BaApar, pinuufi gaHouy (ax napalerorpaM,

to6ro ArBrCrDl

Bi}noai}u:128

eo. no = a 1.s.8 \22

a AB, = K, rouy WC) ^ (ABC)o9=CD,an CD. 2) lAKB - ZCKD rrt eeprurcarsui, IABK ICDK ax suyrpiuui pisnocropoHni. \ABK * tCDK sa ABoMa KyraMr{.

ffi j{fi$i$$

=92'BK

4'5

=ro

{cv). 1) 3o6par:renusu pirno6ivnoi rpaneqii..43CD (BC 1l AD, BC < AD) e goni.nrna rpane\ia ArBrCrDl, y flKoi BlCr l) AtDr BtCl < ArDr ocxi.nrnn npz napaJrerbnoMy npoexuilosaHHi :6epiraorsca naparersuicrs nprMhx ra nponoprliiluicrs si4piarcis

u - AAz,

(APA) n B = -B,-B,, AA2 ll Bp2. a "A{rq a BrBrQ Ba ,qBoMa RyraMu. roar eA,- x*r; =% - ArA,=1. P 1 aA" aA, ++

o4Hiei a6o trapareJrbur.rx np.fi Mr4x, 2) floenauzmo M, e BrC, rarc, rqo @8, =

ArB, = QA1 + 8Br roMy , + 3x = 6; 4x = 6; x = 1,5. QA1= 1,5 BiOnoeiOu: 1,5 cw,

ArB, = 6

3r cu.

sa yrrnoeoro B,Br= CrCr.

ArA,

crvr.

B182ll CtC2, ro aa s,Tacr[eicrro napateJr]'Hilx

nlous,ua A,A, Togi aa oaHaxoro lapareJrorpaMa, ocxi,rsxu trporu:reffi-

ui cropoHn pirni i napa.ne;rnni. ro ArCrCy',z, CrB\82C2,

A1B1BA,

napalaeJrorpaMu. Ix nporra.nerxuicropouu pinHi: ,4rC, = ArCr, B,C, = BrCr, ArB, = ArBr. Touy r-4rBrC, - rArBrCr Ba rpboMa cropouaMrr.

So'u.., =

S*,u..

Bi4noai1u: 6

1l )BtCt 4Bt=t

3 4- 6 (cv').

ctrr2.

26. Ocxi;lrxu npil napaJT eJrr,Hoivry npoemqioeaHHi a6epiraeT tcs nponoprlifinictr, niapicxis oAsiei npnmoi, ro ArK, i KlBt = ltf( : KB = 2 : 3. .- r''B,.2= 75 2 (cu). ToMy,41Kr -' .' *=ti oa Bi4noaidu:6 cyt.

674

BrMr: MrC, = I : 3,

ao6paxeHHfl traparerbhoi npoeHqii trepneHAlxyrspa, npoBeAeHoro B ToqKu cxopzcr:aeruocr ni4omum B Hypcy n,rauiuerpii @anrom: npRua, flrca [poxoAurb qepea cepeAZH[ ocHon rpaneqii, nepneHAu KyrspHa Ao H].rx.

,{ra

M 4o ocnoou AD,

cut,

25. Ocxinrxz o

= 128 (cu,).

cu'z.

29.

Bidnoeidu: 1O cm. 24. ALB. n A',8" = Q, (ApAr\

y ffrdoro nci cropo-

r{acrnxa 3

23. 1) AC a BD

AB 84, nx CDDKAB2

su i aiaroHa,ri pisHi).

S-,u,qq =S*"o = 4S*ou=4

erE (*r,).

(cm).

28. Ocxi:rrnu (ABC\ ll a i AAr ll BB, ll CCr I DDr ro ArADDr, D\DCC. C\CBBr i AABBT traparrerrorpaMr. Togi AD = ArDr, DC = D1C\, BC = BrCr AB = ArBr,

Bidnoai0a: 9J3 cm'.

ArDr.

) cepeAH.fi nillia, MrOr: OrN, = $ ' $. Togi M,O, = 3.r cm, O1N1= 5)c cM. 3t + 5x = 24;

MrN,

::ii: TEOMETPIF EincxiHa O. 9. ra ix. ArasevivHnfi piBeHb i:iiiiiiij;iiiiii

3) floaHavnuo P, flponegemo

cepeguHy

BtCr K1- cepe4nuy-4rD,. flo6y4yerrno npsuy PrKr.

MrN, - PrKr MtNr

q B:+ =!; n,*r, M,C. 3 '

iryxanrafi nepneHAr.nyJ'r,flp,

= 18 : 4 = 4,5 (cu). M,c,=

4,5,

3 = 13,5 (mr).

30. Hexaii AB 4anzil ni4piaorc, Ar i 81 npoexqii roqox A i B Ha n.norquHy c, l) AAr ll BBr. Ilpoee4euo uepes npaui -4-4, i BB, nrorquHy. BoHa reperHe nroqzHy Hexait O

cepeArrHa

BBr, ropi AO o a = C, C e AB.

2) tAA,C -," aOB.C (OBt ll AA).

oz-=!nn.=4:2=2 '2

cv.

AA. A.C 8 r+9 Hexail B,C . x cm, ro.qi Oi,= i,C, 2r 8x-2x+18;6;r=18;r=3. BrC = 3 cm; ArC = ArB, * BrC = I * 3 = 12 (cu). BiOnoeiOa: 12 cm.

illiii{iiiill

TEOMETPI9 EinqHixa O. f,. ra in. Axasewrivxnfr pieens

fri:ir 675

22+

5.9.allON = aNOM = tKOM ca ABoMa lcareraMr4 (OM = ON = OK ' KN = NM = KM, To6ro tKNM * npanrz.nrnr,rfi. 3 .NOM: NMz = NOz + OM2; U.tW = nl"' + "" = "JZ.

31. Ilo6ygyeMo nepepis xy6a nrorqunorc (L4rCr) - uorr,rprrxyrulrrd AATCTC; uepea gri upnui -AA, i ArCr, rqo rrepeTr,rHaroTbcfi

, npoBeAeMo nJrorqr,rHy, BoHa nepeTr,rHae

(ABC) no npauifi.4C, AC ll Apr fl.norqrzna (MNK) neperr{Hae rpaus AATDTD no npanrirl KM (4ai cnilrsi rouxz). Anaroriqqo (MNK) . (ABC) = MN. Il;toquuy (ArBrCr) trJlorquna nepepiay [eperr,IHae no npalrifi

P*xr=vaJi; So*r*=US=+ 42

KF II MN.

5.10. 1) ,{zo. Ni 5.s. PMc = Sa"l, = 5.5J, = 15rE (cu).

Orxe, KM ll AAr, MN ll AC (cepe.que:rini* arADC). Ba osHarcoro napa.ne,rrnocri [Jrorr{r{H

- ^[Ao\ oB' =,ls' - g = ^lz5 +s6= J61 (civr). B tAoc: Ac="[Ao\oc, =E.s'=Jrb-% ="60 =s.E (cnn).

(MNK) ll (AA.C).

2) B aAoB: AB

32. Ocri.nrrcu ro.mu A, B, C i D He Jregcarr, na oAnifi nlounlro6ixni, a anauurr, .nerqvrni, ro npaui -48 i CD

,rcaTb Ha Aesrcrrx [apaJreJrBHr,rx fiJrorqr,rHax

Cuoci6 no6yAoBr4 Tariux rrJrolqrru onncanraft

sa,qaqi

xoxnift

Y napaJreJrbHa

o npaivrr,rx .43

CD, a sna.rrarr, i n;rou{unam o i 0. r{epee upaui AC i AD ra BC i BD, qo reperl4rrarorbcs,

npoBeAeMo

uapaJleJlorpaM.

nroult4H y nPOcToPl

B aAoc:

npanri y npoctopi Haar{Baror}'cs nepneru0urcynapH'tt'Mu, ar"ruo Borru fieper?rnarorrca ni4 [psMraM KyroM. Bnacrunocri neplerrArrrcyJrspnux rrpfiMr{x y rrpocropi 1. {epee 4oni.nrny rovxy npsuoi y npocr:opi MorfiHa upoBecrr'r trepnerrAr4rdyJr.Epuy go irei npamy, 2. ffrcrqo Ani npgrui, cNi nepetr,rnarorrcs, si{noniAuo napa.nelrni ,4aoM nepneHAtrBOHI,T rerft flepneuAr{Ky.napni. 3. rlepea 6ygr-arcy rorrry flpocropy, rqo He HaJre]lcllrr nprMii, Mo]rcHa fipoBecrril

nyJrfipnl,rM [p.EMHM, TO

s.2. A.

Croporra tABrC

5.5. B.

4iarona.nr rpani, rqo uae tpopnry

rcBa?qpara

5.6.

5.7.

AB1 = BrC = nC

Bi}noei}a:

= 4x!2;

Pona"

= 3.

,qo

t; =

i:i{:

reouerplg

(cu).

=.8

4a' = al6 (c*).

^[Ao\ oct = ^ta\ ra'

Joc\ oB' = ,16c\

= o(JB +.u[o +

= o.u4o (c*r).

4a' = a",[g

(c*r).

JG) "..

i ool L Arot Ana-nori.rno AoBoAr{Mo, rqo OO, L BO

L BOr

OOr

2. Iparri Aarroro [psMorcyrnoro napaJrerenine4a fipaMoryrHr,rxra. Croponu nepe..._ a BCrD awaxo4uwo aa reopenoro fli$aropa.

pioy

alz

lzxli

BC! = Bgz + CCI; BC, = @vr).

32Jg

J7* "";

Oncc: = '["\u'; n

BD =

'lEc\

JB"l& + Jg' + +' +'l 4' + 6' = .i6BJb + b + 2Ji5 (crvr). = JE + rlis * "lii =

iii:iiliiiii:j

DC' =

J*;a.

nE * u' + Ju' - r'.

p*",o =

(cM'?)'

Einnnina o. 9. ra in. Ara4ervrirHnft piBeHb

DC, =

p*or ='[o'

Bi1noei}a: E.

676

= 5"t, + 2.,6T (cu).

Ocxiarxi,r 6iuai pe6pa rcy6a nepneir4nryngpni Ao npsMoryrrux, OO, I AO ftoro ocnon, ro AATOTO

5,1

N! 5.?. C'ropona LDTAC = 8J2 c*.

. -(a'E)"'6-o+'" -M,A( 4 i:!

lcnn).

iOt- qenrpra nporurerxnlrx rpanefi,43CD giaroua.neft nna4- rorrrlr [eperr,rHy parin rpauerl. AO = ArOr AO ll Apr rowy AATOTO - rapaJrerrorpaM (nporrz,rexni cropourr pinni i napa.ne.nrrii).

r-t2-

,lrs +86= ,,6T

i ArBtCrDr xy6a

nrzpaxaerrca ue'

S.g.And.noriqso

AC =

BOC: BC =

5,11. Hexafi

pea cropouy aa (popivrynoro a = aJ2.

np.fiMy, nepneH,qHXyJIspHy Ao Aanoi. 4. flrcllo npsnra [ep[eHArr]cyJrspna Ao oAniei B ABox napaJreJrbHilx trpffMrrx i,rexurs B Hr{Mr. B o,qnift uouluni, to soua rreprrenArrxyJrfipna i Ao 4pyroi npauoi'

5.1. A.

"67

4) 3 r,AoB; Au = ,[Ao\ oB'

,[ri

5.4. B.

P*"=5+J3++Jar

5,1 . IlepnengrKynepHicrb npeMt4x y npocropi

5.3. A.

rJz

=,[Ao\ oB' = 17 + t'z = J9+ 16 = 5 (crvr). B dAoC: AC = {Ao' +oe = J"z - rz = Jg -%= J34 (eru). B rBoC: BC = .tje *oB, = ,lt, * = J% + 16 = GT rcm).

nroquHrt.

= 16T +

3) B aAoB: Au

(ACD);y = x4s, MK ll cD; (BCD) n y = NP, NP ll CD, rori MK ll NP. Arra,roriuno, MN I KP. SeiAcr po6uMo Br{cHoBoK: MNPK MOAvnb 5. nEPnEHA]4Kv.IlgPHlCTb nPCMl4X

aBodz BC = JoC\ oB" = ^ts"

P*"

4.15.

Tper* nnorquna

za ymonoro). Togi

lltiiliii:l:ii: TEOMETPI9 Einsxina o. 9. ra

86 + Js + 16 + Jro + eo =

iH. AraAeMi,{nufi

piaenu:j!!: 677

5.13. Poss'saaHus asalotiqHe .4o J$ 5.12. BD = ^,tEC' * CD' = 't"' - b" = J4' - r" = Jro+

c,D -- ,En'

"' =

Jr'z +

= 1i - ea = ,6a = 2r,47 tcm). BC, =

2fi7

KyTHI4rtI4.

Hexaia ABC\ cepeguna AD, K

gart:rir tpanuntuuii rerpae4p. M cePeguna BC.

pieuo6e4peuuir, 6o AK = KD

l) vAKD i nucoru) pionocroponHix

(a sHavurr,

rrJotrll{Hu.

6y,qr,-,sniii np,rmifi, rqo

i npoxo4urr qepea

nepneHAr4r{yJrfpHa

re:xurr y n.lrorqfiHi (.43C)

rorrlcy C. Orxte, CK )- AC.

(cwr)

5,29 . ABCD

npane4ia (BC ll AD , 1A = 9O").

I)(ABC),MeL

pienocropouui rpu-

Ocxiarrn I -L (ABC), ro MB I AB. Ocxiarxu r:panerlia ABC D npavoxyrHur{a 3a yMoBoo, ro

AB I

BC.

Tanunn vlrnom, npaua -AB trepnenAznyJrapHa ABoM np.flMrrM MB BC nrlorrlr4Hu (MBC), a rle osaaqae, rqo,4-B -L (MBC) (e a ymonoro).

stx ne4ianz

we4ia:aa r'A.I{D 3a yMoBoro, a snaqzrb.

MK LAD.

i nlorqlrnn), ToAi npgma CK

oBHaKlr fleprrenAr4]ry-

rpnnyrnraicin '4-BC

cDB.

,rcpnocri np.flMoI

(cu).

TerpaeAp HaarrBaerbc,fi npanunbtlutt, srtrrlo aci itoro rpani

5,14.

c;ri4nou

5.24. HB L(ABC),CK I HB,romyCK L(ABC)(nac.iri.aox a oBHafi r{ nepneH4u rcy,rspnocri npnmoi

vl D

Fc' * cci = ff - "z =,tU - e' = d6+ 64 = Jao = a.6

p*o", = 2J5 + z^,Ei + +'tB = alr +

:",, -tfl ,"

(cul.

'fzo , ccl = i;;t

5.27. AB L a, CD ll AB flK ocHoBr.r rpaneqii. Tozi CD -L o ga sa-

B =

5,3O. ABCD

npsmoxyrnux, roMy

AD I AB.- AK L,(ABC), rovy A]< LAD. flpnma AD nepnenAr{nyrfl pna ABoM tpamuvAK iAB nnoqnuu (,4-BK), nni neperzHalorrca n rovqi A. To4i AD L(ABK) Ba oaHarroxl nepreu,qurcy;rapnocri npfiMol i n,roq[Hz.

eracora:

2) AHa,'roriuHo, rCMB - pieHo6eApesuk,6o MC = MB' v'etiana i eucora: MK L BC.

55.2, flepneHguKytepHicrb npeMoTra nroull4Hll y npocropi

Hffi$i :

/iiu;ri'^..r-.r.tr

i1.i$ffi

flpama, sna treperr4Hae rrJrorqrrHy, rlaar4Baerbcn nepneu1urcynnpnom go qiei n;rorrlrrnr, slcnlo BoHa neptreHArrr{ynffpua Ao 6yAr,-sxoi npauoi, qo 'nbxrzrr' Ha qifi qepes TorJKy ix neperuly. Oasana. flHrqo npama trepneHAZnyJrsprra Ao ABox trp.trMrlx' sri 'nexcatr y n'norqlrri Ta rreperrrxarorbcs, ro BoHa nepneuAttny,'rJlpua go 4aHoi trrolrluHu' HaoriAxu. 1. .flrqo nJrolquHa neplen.qflIdyJlfipHa 4o o4niei 3 ABox rrapaleJrbuzx npffMr{x, To BoHa nepneH.Ilu}tyrfipHa i 4o apyroi npnvoi. 2. ,[si npsui, nepileHAnxyrspni ,4o o4uiei fiJrotr{r'rnu' napareJrbHl'

nrouluHi i npoxo4lrs

5.18. B.

5.21 . B.

5.16. A.

5.19. B.

5.22.8.

5.17. B.

5.20. B.

5.23. B.

5.15.

5.24.

5.25. 3a yMoBoro rrz n, [pt4 Ltrr'oMy rn I cr. AoseAenro, rqo i n I tx' Poar:rsneuo gei npaui c i d' tnoulu]f.u cx, nHi neperzualorbctr' Ocxinr'xn

5.31 , Vci rpani xy6a rcBaAparq, rouy AC L BD. Hexaii M cepe4uuaAB, -noanaqzrro I{ - cepe4zHy-4D. Togi MK cepegaa rri:r.itt IABD: MK ll BD. To4i MK LAC. flpoee4euo MN ll BBr MN LAB. Ilaorqnna nepepiay ne' perrrHae rpanrA,B'C,D, no ni4piany NP, NP )l Bpr MNPK - uyrcaunfi nepepia, npsMonyrrrrn. Il.norqrana repepiay (MNP) napaae,rrua r.norqr.*ri 4iaroHaJrr,Horo nepepiey BBtDrD, ocnimnz npoxoAurb qepea 4ai npaui, ani ueper:zuarorbcfl, flapareJrbHi 4eou npaMuM

nroqnHu BBtDtD.

BBtDp 1,4C. Biaouo, ruo flKUlo trprMa

cr'

ro m L c i m L d'3a oguaxoro. A ocnilsxra n ll m, ro n L c i n .L d' Togi fipflMa z nepneHAuxyrlrpHa ABoM npfiMuM c i d nnoqu:rru cr, rrci neper:zuarorbcfi' Suavurl' n

nepneH.4r{KyJrfipHa o4uiil a aaox napaJrerbHfix trJrorTlfiH, ro BoHa neprreH,qr.rxyrflpna i gpyrifi. Orxe, AC I MNPK. 5.32. Hexafi ABCD raparerbrpaM, BK f (ABC), 5.33. Posn'saanns ana.noriq FD L (ABC). Tosi. BK ll DF. AB CD nn npor:use ao Nh 5.32. .nerxni croponu fiapanerorpaMa. flni npami AB i BK

rpauuu DC 3a oaHanoro (ABK\ ll (DFC\.

lJ'rorrlr4Hn (,43K) napa"remni ri4noei4Ho

i DF nnoryuun (DFC).

(aa osnaroro).

AD iif;tiiitif* rEoMETPlc

Einsxina o. 9. ra ix. Araseuivxui pieenu;!:i:i 629

5.34. Hexafi/BCD - aattufr 4onfurrnrft vorzpr4r{yrgr4r{. AC n BD = Qi BO = OD. M e.(ABC); MB = MD. BD = 6 cu; OC = 4 clu; CD = 5 cv. tBMD * pinao-

A)l=z^tE rE=4"(cm). F) J=4Jt (cM).

6eApesuii, BM = MD 3a yMoBoIo' romy ocxi.nlrcr'r sficora i MeAiaHa d BMD, BO = OD, ro MO

oo=!nn=9=s 22

A) 1=6J2

("*r.

npauoxytnrzft (erranercrrcuft), CO LBD- Taxuuuunou, BD LCO, BD LMO' BD, -rouy BD L (MOC\ 3a ooHar(oro flepnerrAnr{yJrspnocri npauoi i n'noulunra'

aCOD ..L

$ 5.3. IlepneHgltKynep i noxnna. Teopeua npo rpl4 nepneH4llKynepn

Bi4piaor AB AO frJrorqrrHrl

C)(

nepneugtrxyJrflp, nporegenvfi s

t = 4^tE.

+.2 = 8 (cu).

D t=lJr.E=o

5.43. S =AD2; AD = rlS "*. 3a reopeuoro fligaropa /Jl1'?

nox:u.na, BC * upoerqia noxu.rioiAC na nJroulr{Hy o[. ocnosa noxr'rroi' ocnona flepneuArrrcyJr,Epa, C

= 11 (cu). = 16O + rOO = 14 (cu).

D m=Jaa+too =12 (cu).

r1 -=./os+roo=13 (cu). Al m = JffO+fOO = 16 (crvr).

npoBe,4errfix

s o4riei rovrcu 1. Ia r:ovrcr.r, rqo He HaJrexctrrr n.nou4uni, Molrcna rrpoBecrrl oaun i ri.nrnra oAilH fiepneuArrxy.nrp i 6esni'r noxfirrzx. 2. ,{oaxnna trepneuAfircyJrspa Meurna 3a AoB]rcr.lHy 6yAs-sxoi noxnnoi' 3. Pisui noxuai rrnarorr pinni npoenqii i, nanuanu, uoxr.ni, rrlo Malorb pirni npoenqii, pisni iuim co6oro. 4. Bi.nirua noxtrJra uae 6i.nr'ry npoenqilo, 6i.nsnrift upoerqii niguori,4ae 6i'nrura

5.45.

lepnerrAlrxyJrapna

5.36. E) OB < OC.

5.37. E, 5.38.

5.47. y Nb 5.46 6yno gonegeno, u\o a MCB - upauorcyrnui, 3 tABC: AB = 5 cm (erranercrxzft rprzxytrrarc), 3 aACM:

I)1i5.

Lag =

I) MA L AD i MC'L CD sa TTII. 5,41. Bi1noei)a:2J IHAC;4) IBAH;5\ IHCB. 5.4O.

,{iarona.nr KBaiqpara a = "J2

i::::::

5.46, MA L (ABC), AC I BC oa yMoBoro. MC L BC aa TTII. a MCB npauo*yruufi.

i Ao noxu.noi. I nannaxu: axrqo

5.3e. A)

680

BD; 2) B,C;3) DCr.

npffMa, rrpoBeAena qepes ocHoBy [oxI'{JIoi n-a nnqu4zni' nepnerlAr4rcyJr,fipHa Ao noxr4lrol, To BoHa rrepFenAr4rty.nnpna i 4o upoexrlii noxz.noi.

5.42.

Teoperua no rpr{ repreuAurcylapu (TTII) flxr4o npama, flpoBeneHa na nlou4rai qepes ocuoBy noxz.noi, nepneHAl4xyJ'r.fipna 4o 'ii npoenu,ii' ro BoHa

+ MDz'

A) m= vEl +100

lo'lxu A

noxrrJra.

= 1lD2

AM =,[S +to" = Js+ roo ("*).

BracrNrocti reprres,qrftyJrspa i noxra;roi,

(c*).

(crvr).

;

E B

FEOMETPIS Einsxina O.

,tqM'

fi60

Ac' = llr'

4' = v/iz4

+ 16 =

= a.,60 (cm).

3 aABM:

pm =,[ettt, + as, = Jb,

-tr, = Jrb+u4

= JrOS = ra

(cnn),

5.48. 3 ro.mn A Ao nJrorrlr4Hr,r clc npoBeAego rrepnerrAlrttynap AB = 12 cm i noxuly AC. "fozi IABC * npanor<yrnrafi. Hexaft BC = x cur, ro1i AC = (r + 8) cm.

oJz.

3a reopemoro fli@aropa AC'z = AB2

(x+8)2=122*x2i

BC'i

x2+76x+64=L44+x2;

.dcM

l6x=144-64; l6x=8O;x=5. AC=5+8=13(cu). B

ra in. Axagervri'lHrft piBeHb ii:ii:'.:*itii!

BiOnoai0u: LB crc. $iC.t:;ii:il:.ii

rEOMETPI9 Eincxixa O. 9. ra iH. Axa,qeMi,{nraft pieexu

l.{"!

681

5,55, ABCD

5.49. AQ L a, AB i AC - noxz.ni, QB i QC ri4noni4no ix npoexqii, 3 trpffMo(yrHl4x rpuHytHuHin ABQ i ACQ: BQ'1=ARz'AQ?' 1r' =.lf€p -r44 = fs = e 1". BA = {Au" - AA' = 'l:.s' CQ2=AC'-AQ2; ca = ^tls' -rz" = Jr25 - r44= r,Ei = g (cu). BiAnosiAb:5

4iaroHa.refi trapaJreJrorpaMa AO = OC, ro6ro BO me4iaxa i sr,rcora aABC, lfe oanarrae, llqo aABC pinno6egpeHnfi. Tomy ABCD pom6.

* + 6r't

Bidnoeida: 24 cv.

cm.

Bilnoai}t:

+ SO2; (x +

3)2

5.56, Hexaft B roqnr4 A 4o n;rorqranz cr npone4eno 4ri noxuni AB i AC rax, r4o AB : AC = 5 : 6. flponegemo AO L u, roai OB i OC npoeHqii noxr.r.nzx. Bpaxo-

Byroqfi Te, rqo 6i.nrrra noxr4Jra rrrae 6i.nrtry npoenqiro,

uaemo: OB = 4 clu; OC = 3t[ crt. flosHaqfiMo .r xoetbiqienr nponopqiftnocri, To,qi AB = 5x cm, AC = 6x cw. Bupaeumo s rAOB i rAOC gonxczHyAO.' 3 aAOB: AO2 = AB2 - OBz = ('x)z - 4z : 25x2 - 16.

152'

9 = x2 + 225;6x=225 AO = 36 cm.

9;6x =276:':c=36.

AB i OC - noxnn| A.B = 26 cv, OB = 1'O cla, AC = 30 cm. 3 IAOB: AB2 = AO2 l OB2; AO2 = AB2 _ OB2 = 262 _ LO2 = (26 10X26

5.51. AO L a, AB i AC

ix npoenqii.

a .AOC: Ao2 = ACz - oCz = 1ory, - (s"6)' = B6x2 - 27. Ilpupinnaenno orpuuaui BHaqeHHff: 25x' - 16 = 36x2 - 27; 36x2 - 25x2 -- 27 - 16; l1x2 = ll; x2 = li x = 1 (r > 0).

+ 10) =

Orxe, Ao

=16.36=576.

3 IAOC: ACz = AO2 'l OC2; OCz = ACz AO"; oC = ,tTo' - sza = Jr00 - 5?6 = .tm = 18 (ma).

5.57. 3 ro.rxu A

K j' D

y nrroquHi B. KM L c\, ror.i KM xy.nap

ner.<aru

4O;

x = 5.

y n;rorquai a, M i N

Hexaft BC =

OB,

Ocrixrru DN * KM, ro

(s)'

cM, roAi

Ag =

.r)

crr,r.

= 92 , x2 = gI _ x2. = 12' _ (L5 _ x)2 = 144 _ 225 + 3Ox _ xzi 225 | 30x x'z: 3Ox= 162; x =5,4. 9.6 (cm).

5.58. 3 rouxu M, M e o., trpoBeru MO L d, MA

- 37 cv roxu,li. Ocrci;mr*z 6i.nrmiii noxra,riii ni4noei4ae 6i"nrua npoenqia, ro AO - npoenqia noxrz.nor AM, OB npoexqis BM,

(cm).

i MB 13

aa yMoBoro AO : OB = 7

AO=7xcu,OB=rcu.

:1.

3 aAOM: MO2 = AM2 - AO2 = 37' - (7x)2 -- 1369 - 49x'. 3 r BOM: OM2 = BM' - BO' = I32 - x' = 169 x2. 1369

4912

= 169

Mo = Jt6s 4

= v6?6 = 26 (cm).

x2; 4812 = 1369

L69; 48t2 = L2OO; x2 = 25i

= tT6s - % = tT44 =12

(r > 0).

(cm).

Bidnoaidu L2 cn.

Bi1noeidu: 26 c*r.

FEOMETPIS EincHiHa O.

- r44

81 - x2 =144 AC = 75 - 5,4 BiOnoai}o: 9,6 cttr,

Tlpote1emo DC I B, ro4i DC - nepnengutroxr'ura. DC = 5 ct't, DN = 7 cM

npoenqi" u]4!t""u?ry ua n:roquuv

;::::::

,lr2b

BC2; OC, OC2 = oc, = Ao2 _ ACz. OC'

5.54, AC L u, AC L 9, AC = 10 cvr. BD - noxlata. Ilporegeuo BK L F, rotl KD - npoexqia BD Ha nJrorqrruy S. KD = 24 cw. 3 IKBD: BK = AC = 10 cu; BD2 = BK2 i KD2i BD ="tEK\ KD' = ^lTo\ r4' = vmo + 5?6 -

6A2

ao' Jtn'iz'

tcDN: cN =^[DN' a'-UVz -rz =J4s-28=rE4=2J6 Bidnoeido: 2rF cu. B

rypcy n,ranimerpii ni.qouo cninnigHorueuu,fi: Brrcora np,fiMolryrHoro rprrxyrur.rKa, onyuleHa ua rinoreHyay, c,cepe4uim nponoprlifiuum mirx npoexuinun xareris Ha rinoreHyey, a xolrcsufi rcarer cepeAHe uponopqifiHe ui:n rinoreHyeorc i fioro npoexqierc na rinorenyay.

cttr.

B, rovrca

DN/p.

dB'

x2 - 8x + 2O9; 8x = 2O9 - 169; 8r = AM = $6s - b' = "tT6, - % = ^1144 = 12 (cu).

5.53. cx

cL.

= ./ar :'s 1cuy. Onycrziuo a ocHoBZ nepileHArrdyJrflpa O OC L AB. AC npoeHqifl nepreHArinyrapa AO ua noxnly -A-8,

x?.

!69-- x' =

(cM).

Ha nJrorquuy ct npoBeJrr{ nepneuAr4nyJr,fip

AO =12 cu i roxury-4-B = 15 cm. OB noxr.r.noi -48 Ha nJronllrHy - npoeHqis 3 IAOB; AB2 = AO2 * OB2i

= x2-8x+2O9.

Bidnoaidu:

='lrb-' - tG = ^lrb - IG = rb = s

Bi1noei)u: 3 cm.

5.52. MA L a, MB = 13 cira i MC = 15 cru - roxu.ni, AB i AC si4noni4Ho ix nPoenilii. Ocrci,,rr,xn MC . MB, ro AC - 6i,lr,ura npoexqin. W,ttt,. Toruy AB = x cM, AC = (r + 4) cu. !ftffii{:fi;:1:;: 3 IABM: AM2 = BM2 AB2; AM2 = 132 - x2 = t6g 3 yCAM: AM2 = CM2 - AC2; AM2 =I52 (x + 4)2_225 _ x2 _ gx_ 16= flprzpinuaemo orpramaHi snaqeunfi

P*"r=6'4=24(e\c).

AO 5,5O. SO ,L cr, SA - li upoexrlia. - noxtrra, SO = 15 cu,AO = x cv, SA= x * 3. 3 tAOS (ZO = 9O"): SA'z= AO2

i BD fioro 4iaroc a, BD L a, 3a- sracrfisicrm

,fiapa!'rerrorpatr, AC

na.ni. 3a yMoBoro AC

ra iu' AxagevrivHuft

piBeHb

iii:iiili:i;iii

iii{i:i1:ilti: TEOMETPI9 Sinsnixa O.

ra ix. Ara.4eruiqHrfr piBeHb

il*

683

5,59. Hexaft y IABC

= 90". DC L (ABC), DC = 1

BC=2tv,AC=3sNr. AB =

^tW + BC'

"s^..- =

=.,12'

.g = G- I = ts

ttn'

(c*).

I AC. Bc =!.r. t= B (amz). 22

flpone4emo CK L

o.s-- _2 66 _ t s*, = BC; AK =: 12 (cm). )AK11 8 aADK: DK2 = AD2 + AI{2'", AD=.[DK" - AK' =$+" -tz' = Je4-rrlet.trl = J5.4 s6 = 2.6\6 = I z16 (cru).

AB, ropi

1 .S -_=:_4B.CK: 2

2S..^^ 2.3 =_:"q= AB J13 -=F

Bilnoai\a: 12J5

"m +=1=e,r J13 2

"t =|

Bidnoai)a:3,5

Jr3

(gM).

(4ur,).

AD = BD = t AC D

= 44 cut, AC = 37 cu. 5,60. Hexaft y IABC AB V tpuxyrrrury nporra 6i.nruroi cropoxlr Jre)Rlrb 6i.nr'rrruir rYr. rcyr, ro6to A - 6i.nrrrrait AD L (ABC), AD = ht= I c16. Ilpone4erro DK L BC, rogi sa r:eopewrorc flpo rpr4 nepneH-

tiHf,lffi ii$i'ii:iirlfi

3 IADK:

DK2 = AD2

5.61, Hexafi y

BC =

-A'

= !61 + 144 =

,{ri

Meurrror

rrJr

- o\(p -

b)(p

- r)

^60

n.ronlranra,

rlo ileperrruarorr,cg,

HagrrBalorr'cst nep'

orq r.rHr.r nepne n4r,rx y,r.a pn

BJracrfisocri 1) flnqo a rorr&lr,

f,Ix i e nonepe4sift sa,qaqi, cfiopucraeMocb norrrr4peHrrM upuiiolrou: 3uaftAeMo flrorqy rpnr{yrHlr$aABC aa ogHiero $oprrny;roro, a norim a gpyroi @oprrnynra onaiigeuo BHaqeHHfi sucoru AK.

= J p@

=fioo-48 =Jn=zJrg, = !62 -16 = = c (cru).

axrqo rper,ff nJrorrlr{Ha, nepnenAilnyJ.flpHa Ao lpRuoi neperuHy qax nJrouluH. neperunae ix IIo fieprreHArarcyJr,fi pHrrx npfi Mr,rx. Oanarca. Exnlo o4na B ABox nJrorqran [poxoAr4Tb qepe3 npfiMy, nep[eHgrftyJ]spHy ao Apyroi n.rouqr.rnr.r, ro qi

3a yMoBoro.

s*"

cut trpoBeAeMo

nen Ourcynapuu.lLz,

cropourr, rovy ZA - uenrurfi. Ilpoae4eno AD L (ABC) i DK I BC, toAi AI< L BC Ba reopeMolo npo rpl{ fieprTerrAtlxyJr,spu. DK = 24 cw

lin

18' = Js?6 + sr4= v600 = B0 (mr).

$-5-4. Ilepnen,qrarynrpxitTb nroulr,rH- 3anexnicrr Mix napanenusicrp i nepnex4nxynnpHicrxl npcMlrx i nnoulum

(cm).

aABC AB = 2O cru, BC = 11 cin'

Jr4\

- (+J5)'

Bi}noei)a: 6 cn.

V tpnxyrnzrcy rvreumqfi Fcyr JrexrlTr' nporfi

cu;,4C = 10 cu.

3 rAOB: OA= ^tW - OB' =12 (cu).

Au=,171r',-Be=

AK=& 44 BC =ry

repona: o =LL

alg

3 aABC:

AC = 13 cv.

Ba goprvyaoro

clr.

go ftoro nJron{rrur,r nepnerlAnrcyJrsp OA, BD - Aorr4rrHa Ao rcorra, rouy OB I BD, ro,qi AB L BD Ba reopeMOrO npo Tpr{ fiepneH,4rrnyJr,fiprr.

AK2;

DK -- ^[AD" * AK" = Js' Bi1noai}t: 15 cw.

npauoxyrnrafi pinro6eApeuzfi Tpr{rcyruurc,

5-63- B qerrrpa ripyra O psniyca OB = 4

4.8.11 .2 = 264 (c*a2).

.BC; Are -*'- s..-^ -w( =!.a11 2 ---

p= 75+37+44 =+x. , S*" =,[r{n - o[r - b)(n - ") JI6 €.

o" =1.s0 = rs ("*). \22

BiOnoaiOa: 3O

Arrrcyrrt{pf DK L BC, 3a (fopuynoio lepona anaiigeuo nlorqy aABC:

crtr.

CD=AD=18cu. B dcDF; FD =,tEc\ cD'

Au'z.

s ' 11.11 4 =

pinno6e4penrafi. Ocrilr,nra B rpr{KyrHr{Ky lrome 6ytu ri.nrrra- o4uu upamlrrt Hyr, To qe rre Mo)fie 6yrr-r rcyr npra ocnoni. Touy srrqo -4.C = BC, ro lC = 9O'. CF L (ABC), CF = 24 cv. FD L AB, rogi ea reopeMoro npo rpr.r nepneHAlrnyr.ffpu CD L AB, CB.- sucora i ue4iaua a,43C,

= I5 cwr, Bf

ijiui

tr€6

5.62. Hexafi aABC

=,/t.i*l ! lJ13l v

=i*

(AM).

J13 3a reopenoro npo rpn neprreHAhnyJrspn, ocxilsxra DC L (ABC), CK L AB, ro DK L AB--\2 = B .DCK: DK = JcD" +cK" = 13 ^/t*9u

"*,

qo Jrelrrrrb e o4uifi ie nepueu4uxyJrspHHx rrJroulfiH, rrpoBecTr,r fieprreH4r,rxyJrfip Io Apyroi rrJrorqr,rur.r, ro qeft nepnenAr.rnyJrgp reuturr, y uepurift

0(.\0=a,y.a=c, YaB=b,aLb.

nJrorr{r4rrt.

2) flxqo r o4niii s ABox nepfieuArrxyJrfipHlrx nJrorqrru npoBecrra rrepnenAzr{yJrffp ix .nixii reperrHy, ro nin 6y,qe trepnerr,4uxyJrffpnuft 4pyrift n.norqrani.

go 20-

= zz. ='t1'z' 1L s' 8

=.tfLJL r - r., = 11. 3. 2 = 66 (cru'z). 684 i{:l reouerplg Sincxixa o. 9. ra iH. Aralreilri'{Hrfi

piBeHb

5.64. B.

5.65. r.

5.66. E.

5.67. B, A.

ii;}iiiiii:$

[iifii{i{:::: TEOMETPI9 Sincxixa O. 9. ra ix- Axa,qeuivnrafi pieenu iililr 695

5.79. Teopema 8, 5.68.

5.80. Hexaft gsi n.norqunu o i p neperunarorbcfl rro npauifi a i neprer4zxyJr,spui Ao n;rorqunn y. To4i vepes 6y4r-axy rouxy npamor a npoBeAeMo fipflMy, nepleHAlxyrflpHy trJrorrlnHi y. Bi4noxi4llo Ao Br'racrr4Bocri, c@opmylsoeauoi n reoperri g, 4a np,fiMa J'relrcl4Tb i n n.norqrani o, i n n.norquni B, ro6ro a6iraerrcs a npJrMoro a. Snigcz a Iy,

L (ACD),'roMy BM L MD' BM BD =,IEBM, = ^lr.2b = rrE (cu).

5.69.

(.4-BC)

5,81.3a

MD'

AC = BCJ|= Bidnoeidu: B.

s.7os.71.

s.E (c.). AN - rqe * NC = Jlo.%

* PC' = 16" -

"lzr

= rJE (cu).

(cm).

Asaroriqno Moxtna AoBecrx, ryo i a L

cx, i B nepnen4rHynapni. CD npnua neperfiHy rrJrorqu:n. AB = 25 cm 4aHufi ni4pisox, A e 9, - rovnr,A Ao nrrorrlrux o, B e s. AC L CD nigcranr, nig

CB i AD

5.73. B.

roai o

rcDK:

="ED\

DK" =

'lz'lp

npoexqii ei4piana

HHcriB. 3 rABC: AB2 = AC2 LBC2;

1l <p.

nEz =

-A-B

ni4norigno Ha nrrotqu-

BC = ^lqnz - de; BC =.JEs" J, = J6% - 49 = G?6 = 24 3 r ADB: AB2 =..AD2 i BD2; AD = J AB' - BD" ; Ap = J2b' - tb" = .JE2b - 22b = G00 - 20 (cm).

5,75. Ocxim,xu ABCD - HBaApar' ro CD L AD' Ba yMoBoro (ABC) L (ADK), DK c (ADK\' ro*rv CD L DK' 3 yADK: DK2 = AD2 + AK2 - 2' + 22 = 8'(cM)' B

AC=7cw. BD I CD ni4craHs ei4 rovxu B Ao n.noulunz 9, - 3a BD = 15 cu. reopeuon 8 AC L CB, BD I.4,D. Toai

yMoBoIo cr

5.82. Il.norquHu

5.72. KN = :.' AC = 6 cv. MN ,[MK'-KN" =VE'z+6' =lo (cv)'

5.74.3a

i c. f,Irrqo trror{nHv c i B neperunarorr,c,r no flpfiv'iir b, ay I cr i y I B, ro b I y. A ocni;rsnu npaua b nepneH,4r4nyr'rffpna n.norquni y, To BoHa neprreHAunyrspHa 6y4r-axifi npauifi nrorqnurr y, slca npoxoAfiT6 qepea rx rouny rreperrrHy. Taxzm uuuou, b I a i b L c.

Bidnoaida: B,

BidnoaiOa:

rrpfiMrax a

=.tW - Ae : d00 -36 = nGZ = s (cM).

BP ,E(

o, P i y rolapuo nepneH,qnnyrrspB. Ilror4rzna y treperuHae qi rrorqNuu no

yMoBoro nJrotr{u}ru

ui, PoarrsueMo

Bi)noai0t: B,

'12;;v,"

moAy.ns 5.

zJs t"*\.

BidnociOv: 2O av

(crvr).

i 24 av.

Ba,qa.{a, o6epnena go N! 5.82. flnorqnHN o, i B nepneH4uHylapHi. CD - npama neperuHy [Jrorrfrzn. AB = 25 cm ganrfi rigBisoH, A e $, B e a. AC L CD -ei4cranr ni,4 rovxu A 4o n;rotluaz o. To4i - npoexqir A-B Ha nrorquuy a. BD L CD CB = 24 cm ri4craxr, ei4- rounrz B 4o n,rou4unrn 0. AD - 20 crr npoexqia.4,B Ha nrorquny P. 3a reopeMoro 8 AC L CB, BD LAD.

5.83. 5.76. 3a yMoBoro SB L AB, SB L BC, AB a BC = B' ro4i aa oaxaHoro neprreuAr'r(yr.fipnoc1i nparroi i nlorqunu SB L (ABC). fhorquua (5-419) npoxo.qurb r{epea npamy SB, rouy

(s/B) 1(lBc).

3 IABC: AB2 = AC2 i

AD 1A:aO AABC, NA LAB, MC IAC, NA II MC' BiAolro, qo srcrr{o oAHa 3 ABox napaJleJrr'Hr{x fipsMrx nepneHAr{nyJrspna 4o rperloi npauoi, ro i gpyra npaua nepneHArztcyr.ffpua 4o uei. Touy NA L AC, MC L BC' Koxna a nprmzx,4N i MC tepneugunyJlflpna uouluui (,4-BC). Toti trroquHa (ANMC), rqo trpoxoArrr, qepe3 npaui,4N L MC, neprl'enqrrlcylaspua nrorquui (ABC) (re' opema 7, uo4y;rs 5).

AC =

5,77.HEXAft

nepnetAr'rrcyJrsp Ao flJrort\vru ^ABC, 5.78, Hexaft MC MD L AB. Togi aa reoperrom npo rprr [epnel{Aur{yJTrpr4

3 yADB: AB2 = AD2 i BD2; =

5.3 = 15 (cm).

Bi1noei1u: 7 crt

Tecrr 4nn

9acrrsa 2. A.

5. A.

3. A.

6. B.

TEOMETPIS Eincxira O. 9. ra ix. Aragevri'{Hvti piBeHb iiiii:iiiiiii;:ii

[t:iiitiliii

caMoKoHTponK)

vepea npamy MC

i:::::

- r4\es. %)- Ge = T (cu).

cru.

(MC L (ABC)), ro 3a oor{ar{orl trepneHAnrcyJr,flpnocri u:roquu (MCD\ r @BC).

68O

BD=.tqn, An, =J;s, ,0, = [rb-roteb+ror--,li.4E =!669

4. B.

Ocxirrxu tnoryuna (MCD) npoxo4zrr

BC2i

JAa' - BC' = xEb' - 14' = l2b

DC LA.B.

FEOMETPI9 Einsnina O. 9. ra ix. AxaAervri.{xurfi pieexu

iii;:

6A7

8. oB

=.t@4

Bi4noaido: .1.

"61-

s6 =

Gr = s.6

ctuna2

(cu).

17. Y aABC

ACz = ABz

AC2;

20 = 16.

= 9O', rosi AB2 = BCz BC2 = a"

-(2"!i)"

= B6

Y IAMC ZC = 9O" (MC L (ABC), a sgaqurb, MC

* MC2; AM=JAc'*Mc" =Jifl.9 =JE=s

AJ\I2

AB).

AC2

@ttr).

Bidnoai1u: 5 clu.

18, V nprnoxyrnuxy MNKL ML = 6 cu, Ktr = 8 cu, giarona;reii. O - roqxa neperr4ny SO L (MNn, rosi SO I MO.

10. B.

3 aMKL MK = JML" + KL'

rpaneqia. 11. ABCD - lrp,fiMoxyrHa Ilpone4euo AK ll BD, ABDK * npaMOKYTHI]IH.

12. SB L (ABC), touy SB 1,43.

tn = ,!i AS

AK=BD,CK=CD_KD=

cw.

=.,tEB\ AB'

Bi1noaida: B.

=cD-AB=17-9=8(cm). 3 IACK: AK = ^[Ae 4K' ='n0' - #

'En

(crur).

= J5o + 04 =

Jioo

Mo =:MK 22

=+=

3 aMOS: MS2 =

SO2

Jrg'

-f

19. ABCD * rcoagpar, AB -- 2 cv'. MA = MA L (ABC\. Toai MA L AC. AC =

ABJ,

2J2

6cu

MO2;

so -*,[MS' - Mo'; so

Bi1noei}u: B.

(crvr).

Bidnoai4a: 12 cw.

= 6(cru).

=1F -8"

(cna).

JL6g

- % = JI44 = 12

(cu).

cwr,

rcv).

3 IAMC: MC2 = AMz 't AC2;

MC=.{AM\ Ae,

A Ji"*

rvc =

1lr

+(zJi)' ="4.s

BidnoaiOu: 3

13.

BD = 2Ji cvt; BD, =

Bi0noeida: B.

14.

"[En"

2O. Hexaft ABCD rna4par. SB r @BC), BC- L CD, rorvry SC

I CD '34 TeopeMoro trpo Tpr{ neptreHArrKyJr.ffpfi, 3 SCD (zC = go'lt sD'? -'sc'? + CD'?; " CD = AB = 12 cnn.

n4 = J8.7 = Jn = 2Jd (cwr).

sD = ,lic' * cD'

7, 3, 4.

13, AO2 = AB2 - OB2 = L|'z = 225 * 8L = 144;

oc = ^lqe - Ao' = Jt og

L44

Bi1noei1a: B.

16. A.I<2 = AT2 _ KT2 = !32 _ I22 = = 169 - 144 = 25.

92 =

=,lls'-t++

= J25 = 5 (cu).

Ka = ^[Aa'

J4e

- AK' = Ji" -%

-% = J% = 2G

Bi0noaidu:

= Jg = s rc').

cryt.

(cu).

[,. A

= ./16\ rr, = ,lrs6 -

144 =

= Goo = 2o (cu). Bi6noei0n: 2O cv21

. KO L a, KO * nepreHAnKyl.:tp, KM

22. MB L d, MB fiepilenArarcyJrsp Ao rrJrorqr{Hr4 g, MA noxu:;.a, -AB 'i\ npoexqia, MA = 17 ctr'. Hexafi MB = r cu, ro,qi A.B = (r + ?) crvr. 3 aABM: AMz = AB2 + BMj; 172 -- (x + 7)2 )- x2; 289 -- x2 + lhc + 49 r x2; 2x2 i- L4x + 49 - 289 = 0; 2x'z + l4x - 24A = O; x2 -t 7x - 120 = 0; D -- b2 - 4ac = 4O + 480 = 529 = 232;

688

ii::: TEOMETPIS EincxiHa O. 9. ra in. Ara4euirHri piBeHb :iiii:liiifii:

:ililt{ili:i

A--

roxr{rra,

OM = lO cru ii npoenqia. (x Hexaft KO = , cM, roAi KM = + 2) crvr. 3 aKOM (ZO = 9O"): KM2 = KO2 + aM2; (t + 2)2 = x2 -t IO2; x2 + 4x + 4 = x2 + lO0; 4x = 96; x = 24. KM = 24 + 2= 26 (cu). Bi}noai}a: 26 cv.

/oh

a/ /

a\ /"h

TEOMETPIS EincxiHa O. 9. ra ix" Ara4enairxrafi pieexr

"/ l{:i:

689

t 23 -7 -23 rz 2 =8; AB=8+7=15(cm).

27. aABC

pieHocroponniir, AO - uegiana. OD L (ABC), ropl AD noxnJ'ra, AO .._ npoexqis noxuto\. AO uegiaHa, a Briaqlrrb, i nucora rABC.

xr= -7

Bir)noeida: L5 cw.

B aAOB: Oa =!aC= ] a- 4 2L

cropoHa pinnoctoponnix rpurcyr- cni.nrsa ttut<is MKN i MNF.3a ynronoro (MNK) L (MNF).

23, MN

28, aABC

Bidnoeidu: 8 cv.

2sJ,

Ao .22=! AC = 1.uu

tlr.*a.-..,..-. Frtr:.0:--r-r^.^.-.rr-'

filkft;i:::i:::::::

tiffil#i::i.-ii::il #ffii:ii:i:i:ii:i:i rJtFrs-r-.r+.i....-.' ',i iFl.'ii:::.-,-..:..L

= ^[tJn' - Ao' = Bidnoeida: 45 cttt.

roti

J*",g'z

r{acrrxa 3

{53 -r8X53- r8) = -85€1

noxuri, npoee4eHi - lr.i cr. flpoBe,4eMo AO I oc, -ro4i OB i

25. }JexairAB iAC

5'9

(mr).

OB = 27 ctt, OC = 15 cu. Ocxi,rr'xz 6iasura npoexqia niAnoei4ae 6i:muifi uoxr.r.nifi, ro AB > AC. Hexait AB = r cv, ro4i AC - (x 6) cnr. 3 rAOB: AO2 = AB2 - OB2 = x2 - 272. B \AOC: AO2 =AC2 - OC2 = (x - 6)" 152. Maeuo pinnnuas: x2 - 272 = (x - 6)2 - L52; x2 _ 729 = x2 - l2x + 36 - 225i L2x = 540; x = 45.

-zn

= J4b'

-?r, = J2o25Jrs - Jrzgo

or*te, OB = I ev. Ao = l{6L BilnoaiTu:

690

li:::::

nepneagzxynapni.

q ..\

30. 3a yuonoro (x LF, KT risis treperrHy n,'rorluH o - L KT aa uacrueictro nepi B, rorvry MK I KT i NT IIroquH. 3 rKMT: KM2 = MTz - KTt'z. 3 aKMN: MN'= KM2 * KN2. MN2 = MT2 - KT2 + MNz.3a yMoBoro MT = a, KN - b, KT = c,ro4i MN = tE +b, - c,. leHArrnyJr.fi pH14x

= se (cu).

Ao tr,aoqr,tut4 o trpoBe.qeHo nepneuAunyJrsp AO qux noxunux. i AC - 8 cv. OB i OC - npoercqii Ocri.nrxz AB < AC, ro i OB < OC. Hexaii OB = , cM, roAi OC = 7x cv. 3 aAOBt AO2' = AB2 _ OB2 = 42 _ x2 = 16._ x2. B \AOC: AO2 = ACz - OC2 = gz - (7x\2 - 64 - 49x2. Pisssrrug: 16 - x2 = 64 - 49x2; 48x2 - 64 16; 48x2 = 48: x2 = Li x = 1 (r > 0).

Hq "niHia neperuuy trroquH, rovy AH L HQ i Bq-I HQ (reopema 8).

ro4i .43 ,E , b" . r'.

Bi1noei)o: 36 cttt.

26, 3 roqxn A

29. fl.notlurru

3 rABH: A-82 = AlI2 r BH2. 3 aHBQ: BH'= BQ2 * HQ2. Toai AB2 = AI{2 'r BQ' + HQ'. 3a ymoeoro AH - a, BQ = b, HQ = c,

s rovxuA 4o npoexOC

qii noxzrux AB i -4C.

Ao -,17

ocHoBa,

= Jszo. 286 = Jr?6 = 24 (cyt\.

28 (cu).

OM2;

trJrorqr,rHli

c.u. BK - megiana, orxe, i Brcora. fc = ! dC = 12 ctvt; 2 BKz = BC2 - KC2 -_ 20' 122 = 4OO - 144 = 256. 3 a SBK: SK' = SB'z + BK2; SK = ,IEA' * nf';

.,8 = 56 (cu);

Ocxi.nsxu MO L (ABC), ro MO L AO, a AIVIO npsmoxyrHufi.

AMz = AO2

BS = 8!6

rorrfta reperrtuy 4iarona;refi rcnagpata,4-BCD,

ABJ,

Y a KBC:

OM L {ABC\. AM = 53 ctv-, AB = 2aJ2 cm. AC =

pinHo6e4penwfr, AC

AB = BC-= 20 cru. BS L (ABC), to,qi BS -I BK,

cnn.

oA = ,[AB' - oB' = ^18"4 = vC4 16 = J4B = aJ5 (cu). 3 IAOF: AD2 = ODz + AO2; AD -- joD'z + Ao' = + ae= fi6 - 48 = Joa = s lcrvry. ^[+"+" Bidnoaida: 8 ctr.

KC L MN. FC L MN, KC = FC = 4n!i cm. 3a reopeuoro I KC L (MNF), ro4i ea oeua'rennau neprreHAr4nyJr,fipnoi npauoi Ao rrJrorr{rurl KC LCF. 3 rKCF (lC = 9O'): KF2 = KC' i CFz; ta.z= 2.4 = 8 (cm). KF = JrKC' =

24. O

(cm); :48 = 8

,4ei

noxuli: AB = 4

31. To.rHuA i B rerxars no pisHi 6oru ni,q cttt'

rrJrorrIzHu o. AC L a, BD I s. AC = 3O cu, BD = 18 cu. ,flxulo ani npaMi nepnen4uHy.napHi 4o 4auoi IrJrorquHn, ro BoHr4 naparemui: AB 1l BD, IhouluHa, fllta rrpoxo.4rrrr, qepes napaJreJrbni npnrrni AC i BD, neperuuae trJrorqvuy c no npauifi CD, a ni4piaox AB r rouqi O. rAOC *-aBOD sa ABoMa r{yraMz, 6o ZAOC - IBOD fiH Beprrr{aJrbHi, a tAOC i rBOD - npnuorcyrni. IlosHa.rrrvro CO = x cm, ro4i OD = (16 - r) cM.

(cu).

e ro4i6uocri rpuxyrnraxi.e BurrxBae nponopqifinicrr, ri4loni4uux cropin:

To4i

r[E "*.

rEOMETPIq Eingxina O. 9- ra in. Arag,euisHerfi piBeHb

iiiii::-..':iii:i:i:

iili:iiil{ii{

TEOMETPI9 EinqxiHa O. 9. ra iu. Ara.qerurirnrafi pieexu

iti:

691

AC BD 30 18 #= #, ; = #;t

- rl

30(16

= !8x;480

CO = 10 cm.

B rAoc: Ao =,[AC'*oct =Jsa".Lo' OD=16-10=6(cu).

BoD: oB = .{aD\ BD' -- Je ;W = J:.:,6 AB = Ao + oB = 10t0 + 6./i0 = 16d0 (cn). Bilnoeila: 16f0 cu. rreperrtrry rrJrorqr{rr rpaneqiir4SCD

- flp.rMa *yrnurta ABC. IABC

30r = r8x; 48x= 480;

r = 10'

{500+100 = v{000 - r0",[0 (cu).

32. t

3% = !660 = 6Jr0 (cu).

6.6. r, A.

6.5. F, r" A.

6.8.

i rpr-

npauoxyrnzft, ocxi.n*u nzronyerrcs pir-

sicrr, AMz = AB2

MB2, lDz = 82 * 62, 100 = 100.

IABC = 90", to6ro MB L t. ToAi grs roro, rqo6 rrrrou{rrga (CBM\ 6vna reptren-

Ar,rKyJrlrprroro 4o npnuoi J, norpi6no, qo6 6y.nN nepneu,qlrxyJrspHltulr nparvri I i CB (aa osuaxoro treprreuAsryrgpsocri nparvroi i rrJrorqusr I L (CBM\, s$r\o

tLBMlILBC).

Orrxe, I L (CBM) 3a yMoBn, qo rpaneqii -43CD

6.9.8) /ABO.

6.10. A) ZBDM.

6.12. B) 10Jd crl.

6.13.

6.1 5.

npruonyrua, ZB = 94"'

40 c*r.

A) flpsmoHyrnraft. 6.16. E) 6-19. B,

6.1 8. B)

6.11. B) 12 cn. 6.14. B) LMBC. 6.17. B)

6.20. B.

,(,ra nrnanauenna ryrin y npocropi ra poee'asynanu,fl BaAarr e,qiei re*lra neo6xi,qHo cHarx osnaqeHH,fi rpr,rronoMerprrruNx (pyuxqifi y npsMoryruoMy rpariyrrrlrfty ra ta6.nrzqro ix ana.{eur.

MOAynb 6. Kvrt4 I BIACTAHI v nPocroPl

Cuny c on rocrporo Kyra B rrpfiMor{}TrroMy rprrrcyrtl4riy Easr,rBaerbcr

BiAnoruenn.s fiporr{Jretrcuoro ftarera Ao rinor:enyav:

$6.1. Kyrnynpocropi

sina=9: sinB=-b. cc [dociayc on rocrporo Kyra np,fiMonyruoro rprrr{yrur,rna Haar4Baerbcs si,qsorleHHs rrpnJrerJroro ]carera Ao rinorenyeu:

ffi,ffi

cosg = I{ymon

nix

nptLtrtuJttu Ea3ur-

ia rtyTis, yrBopeur4x nPu nePerrni qr{x np,EMrrx.

Tatteer+cou

BaeTr,cg nrenmr.rft

b^a -; cosp = -.

locrporo Kyra [psMoryruoro rprrryrnr4rda

Itymon nixtt rcyr

uix

uno6iilcHunu npflltumu sasr'rBaerbcfi

npsMr{Mn,

axi neperuuarorr,ca i

teo=!; 'bu

uapa.near'-

ri ganuu nr,rlto6irxnuu [pgMr{M. srn

te1=L.

30"

cos fI

quEy.

aozpannun KA nt'ow uasfiBaetr'cr @irypa, yrBopena 4noua ninn.roulr{HaMll pasoM si cnimuoro fip.rMoro, u1o ix o6mexcye. Bosa naarasaettca pe6pom 1aoepau'' noao nyma (m). .flrrqo s 9 = m' a r m, a c a, b L m, b c F, ro Z(a, b\ * ^ fi:ri\ftrtuia fiyr ABorparruoro Kyra l(a, $.

692

6.2. E) 90".

6.3, l) 6 cu.

i:.; reouefplg EincHiHa O. 9. ra ix' Ara,4euisHrfi

piBeHb

iiiil:{iitli

ctg c

"E 2

^,8

ar6

.i5

t; I

IIde 4enrci OopMyrur: sinz

6.1.f) a = 90".

60'

tga

45'

Idym nitt np,nMon i n,Lour'uHon * \e nyr rvri*c rrpgMoro ra ii npoerc4iero Ha rlro rrJro-

rraorrBaerbc.ff

npoTr4rex(soro RaTeTa,qo ilprrJrerJroro:

o + cos'o = 1;

sin a :::-: cos

= 1961;

sin(90'- o) = cos cr; cos(90'- o) = sin

:::ititlli:i:{ TEOMETPIf, SinsHixa O. fl. ra iu. Araseruirxrafi pieenu

cr.

ill

699

Ocxi,rsHz SO L (ABC), ro nprrMa SO nepnen4r.rxyrflpHa 4o 6y4r,-anoi npnirnoi n n.norquni (ABC), qo rpoxo.4srb qepec roqxy O: SO I KO, SO L LO, SO I MO,

6,21.AO L u, AB - roxrJra, OB - npoexqifi roxuror uryranrzii ryr uirr .A.B sa n.nourzuy cx,. To.qi IABO

loxrrJloro

n.norquHoro

ctc.

= aOLS = yOMS = a ONS aa ABoMa r{areraMz (OS cnilruurl). B pinHocri rpzxyrnrHia BunJrvBae pinnicrr ni4noni4uux t<yria: ZSKO = ISLO = = ZSMO = ISNO. To6ro noxra;ri Sf, St, SM i SN saxuresi go nnorlnnu ABCD

cos.lABo=98 = =1. AB122

tr1A OAHar(OBrrM nyTOM

LABO = 6O".

- a. IAKO

/'\rl

LE = Jr,

IAKO = 60". 6.29, AO L u, AB = 4JE c* i AC - noxutti, BO i CO ix npoexqii. IABO = 60'- ryr ni:+r npauoro-4J3 i o, xyr nrix AC i n.norqznoro o. IACO = 30" 3 IAOB: AO = AB ' sin .IABO = 4rE ' sin 60" =

=,tqF * DC' - JB' r 4' = Js+ro = b (cv). 3 aAMC: AM = ACLg6O" = SJI tcru). AC

Bi|noai\t: 5JF cu.

rpr4nyrHrr{, AC c ct, B e cr, BD L o. To,qi 1-B noxlr.ni, a AD t CD ni4noni4uo ix npoenqii, xyru BAD i BCD - xyru, arci yrBoprororb noxz.lri a nJTotqt4Horo c{,.

piruocroponniii

6,28, Hexafi ABC

i BC

IlosHaqzuo cropony aABC x. tACD nparroHyrnnfi sa yMoBoro i pinno6e4peHzfi, ocHirrnq,A-D = DC stt< npoexqii pieuux noxv.nzx.4B

=4JB

3 aACD:

A_____-

AO = 5.tE cu, Ol( = 5 cu.

6.27. ABCD trpsMonyrHr.rx, MA I (ABC), MC noxura, AC - ii npoenqia. "logi IMCA = 60' - rcyr l.ti*t noxnloto i n;rorqr.rnoro npfl MoltyrHr{Ka.

npoerqig noxuloi - mixt noxutroto A-I{ xyr

roxuJra, OK

-4J{ Ha n.norqzny i n.nouqunoro ct.

z,qxo tE'oK5

No.

a O,KS

6,22. AO L a, AK

AB = L2 ciu, OB = 6 cm.

i BC.

==6(mr). AO6 3 " AOC: AC = "" = :, sin30' I

3 a BCD: BD =

Bidnoeida: 12 cv.

IACD:ACz =AD2

CDz;2CD2 =

,lBc, cD' =

Ocni,rsnrz BD = CD,

pinni noxu.ni. Ocxi.nrrcu no6.24. AO L a, AB = AC xnni pieui, ro i ix upoexqii pinni: OB = OC' Togi rABO = IACO Ba rpboMa croponauu. 3 pinnocri rpraxyrnuHin BrrnruBae: IABO = 1ACO.

ro r

x"; CO=j;.

F:-- = l;' ,

BCD

IBCD = 45". IABD = aCBD tBAD = 45".

npRnnoxyruufi pinHo6e4penuft

Ba rpboMa cropouaMt4 (a6o ea

ocnoBoro BC, ro4i

xarerou i rinoreuyeoro),

I a, AB = AC = 3J, cm noxuri , OB = - = 9O". OC ix npoenqii, IBAC = 6O", IBOC pinno6e4penafi IABC s ryrou npz eeptruni 60', ro4i ir xyru npu ocHoei BC 4opinHrororr 60", piruocropoH nik. BC = AR = Syli cn. a a ABC B IBOC: BO'i OC'z= BC2;2802 = BC2;2802 = L8; BO2 =9; BO - 3 cu (BO > 0).

6.29. AO

M 6,25, O ..._ roqrta fleperuny 4iaroua"reii xsagparaABCD, MO L (ABC). ,{iaroHari xnagpara pisHi i to'moro neperrny gi.nxrr,-

12 (cu).

c,s sanni.n: AO -- BO = CO = DO.

Tpltcyrrt:r't<uAOM, BOM, COM i DOM pinni aa 4soMa KareraMu (MO - cni.nrHllfi). 3 pienocri rpzxyrnrzria:

A-BO:

Ao=JAB"

ZMAO=IMBO=IMCO=IMDO.

Bo"

Bidnoai)u: 3 cm.

=t8-9=16=e

(cu). i

6.30.AO Ltt, ABiAC ro'rna reperrrny fioro 4iar{BaApar, O ronaaeii. SO L (ABC). S1(, Sr, SM, SN - noxz.ni; KO, NO, MO, NO ix npoeritrii; K, L, M' N - cepeAlrHu cropin xna4para. gr[x cropoHu Aortrrtarorbcfl Ao B e ro.rxamrr, Bonu Btr[canoro noJra 3 qeHTpoM y rorrqi O. Touy OK = OL = OM : OM r't< pa,4iycr4 BnlrcaHoto u tragpar,4-BCD t<ota.

6.26, ABCD

694

i:::i: TEOMETPIS

Einsxina O- 9. ra ix. Axa4eruivHufi piBeHb iil;ii;iiiii;::i

npoexqii.

OB = OC = 3 cm ix - noxuri, ZBAC = 60", ZBOC = 90". OcHinHn -npo-

ercqii noxz.nux pirni, ro i cavi noxn,ri pinui. 3 aBOC: BC2 = BO' i OC2; BC'z- 3'? * 3'? = 18. pieHo6e4penzit B KyroM npr.r nepuuHi 60', LABC a orrfie, pirHocropoHHifi: AB2 = BC2 = 18. 3 rAOB: AB2 = F'02 * AO'z; AO2 =AB2 BO";

.ao = JAB'

Bo2

Bi0noei6u: 3 cv.

="{s - g -

iiiilfi:t:i:lt rEOMETPI9

Jg =

EinqHiHa O.

(cv).

ra in. Arasenivxnr pieexu

l!:i:

695

r rpr4ryr:g.vt1, AM L BC - ftoro nrco- AaHrrit ra, AM =6.,6 cu. BCca,Aes,AMIa.AMLBC

6.31 . ABC

aa yMoaoro,

3 |ABC: ABz = AC2 + AB = J6' + 6' =

rogi MH L BC sa reopeMor) rrpo rpr trep-

negArrrcyJrspfi. IAMH = 6O" *

aBHC

.tiHifiuraii rcyr ABorpaH-

r{oro KyTa.

V3=g (crr). 3 aAMH: AII = AM.sin60'= 6JE 2

Bi1noei}a:

cttr.

pieno6e4penurl i np.auoxyrnr,ril: oABC AC = BC, ZACB = 90", AB = 20 cu. AB.(1., C e cx, CM I cr, roAi CM L6ya;-stxifi

6.32,

npruifi

o, qo [poxoArrrb qepe3 rovxy M. Ilponegeuo CK L AB, rogi MK L AB sa reopeMoro nyr nrirx rpo rpr,r neprrenAr{Kyrrfipu. ZCKM = 30" u.norqrani

i a. i nre,qiana ar43C. 'foai KB==AB= 20:2-10 (cu). n.norqnnanrz {ABC)

nucora 1

Bracora CK gir.*rrl,

CK=KB=10cm.

IABC na 4na piano6e4peui npsuorcyrni rpzxyr:uvxn:

3 aCKM: Ctvt.2=!XC =

cu (sx Karer, rr{o reficrrrb rporr,r r{yra.30o),

pinnocroponnifi , AB c a, C e d, CO L a * nepneEAr.rRyJr.flp Ao rrJrorqr{Ha Ct. ToAi croponz AC i BD * noxu.ni, a niApierdn AO i BO - ix npoercqii rra nJrorqr{Hy d.

6.33. alBC

Ilpore4euo CK LAB CK = 4JB cvr.

Br^coay

^ABC,

I AB Ba reopeMoro npo rpr{ nepnellArzxyJrffpfi, i - nyxar.uit xyr uix flJrorqurraMn rABC a. s rCKO: sinICKO=co ='J2 =L. 1CKO=3r,. cK 4J3 2 OK

ZCKO

Bi0noei1a:3O'.

i a. 3G cM;

nJrorrlunaMu xsagpara ABCD

sin tBAo =

cvt, BO =

L; =*

tBAo =

(crvr).

=* 2 =aJz

BC.sin45.

.ABH:

sinzBAtr =

Bi0noei6t 3O'.

BH'

orx<e,

BH L CH), Ocxi,rsrcu ZBCH = 45', ro

(crvr).

s:lZ

AB= 6"12= 2 . znu+

= 2,o..

6.36.

a-4-BC = 9O'), AC = BC. - nprMoxyruklit (ZC AC c s., B e a. BH I cr uepnenpurcynap, BC i AB * roxr,uri,. CH i AH npoexqii rra rrrro- ixaxraft qrrny d. IBCH = 45" yr:noproe xaxyr, ret BC s trroqzuoro u,- IBAII xyr mirx rinoTeHygoro

n.norqranoo o.

Y rABC AB = 30

cm, AB2

=AC'I

BC2;

AB2 = 2BC2; 2BC2 = 3O2; 2BC2 = 9O0:' BC2

= 450;

=LI\E (or) (BC > 0).

'BCH BH = BC.sin4b. =|SJZ Y = rb t.*1. BH 15 1. B aABH sinZBAII tBAt{ = Bo.. = = = 'A8302 3

Bi0noai1o:30', 15 cu. o. i B fieperlrHarorbcfi ,Aeu,BeB,AB=25ctl.

6.37. Ilnorquu,r

no npamifi /,

AC L I, A9 = 5 cm; BD L l, BD = 8 cn. flporegemo CK LIi KB ll L To.qi IACK - nilt\it-o, ulrfi xyr ABorpaHrroro ryra uixr trJroulr,rHaMr.r

iF. ztcK=60..

KCDB -- upfiMoxyrtr4n: KCDB - fiapaJreJrorpaM, 6o CK I DB, KB ll CD aa no6yAonorc, ICDB = 9O' , rony KCDB * npaqorcyruiax.

Orxe,CK=DB=8cm. Ba reopeMoro rcocranycie:

AK2 = AC2 + CKz

fr 411 =,ls'

2AC ' CK . cos 60o;

!2 +8'z -2 s

- z (cm). 6o AC.L I1 KC L l, rouy AK I BK. B .AKB: KB= JAB, - AK" =,1;b" -f = J625-49 = Jb?6 = 24 (cnr). = J25+66+-ao = Jag

fllorquna (ACK) L I,

CD=KB=24cm. Bi0noei1u: 24 cyr. 6o".

BiOnoeida:6O".

(lC = 9O"), - np,nMonyrrrrft AC=BC=6cu.ACca.,Bea. noTo4i BC iAB BH La - cL. - ineptregArrr{yJrxp. AH - ix npoexqii Ha nJ'IouluHy xrtli, CH IBCH = 45'- Hyr, ni4 arrau rater BC Haxr4Jrenraft 4o n.norqrurz. a, ZBAII * ryr rrix ri[oreuyaoo i n.norqrnnoro o,

6,35. a-4BC

696

"0.86

nparroryrnufi (BH L cr,

3 aACK

6.34. ABCD * r{Ba,4par, AD c a, B e d, BO L a. BA - noxuta, OA - Ii npoexrlia Ha rrJrouluHy o, BA L AD src cropoua KBa,4para, OA LAD Ba reopeMoro rrpo rprr nepfieH4urcyJr.flpz. Togi ZBAO * ryr ui*c

- Y aABO: AB = 6

BCzi en=JdC\BC";

= 6.uE

ii:i: FEOMETPI9 Sinsuixa O. 9. ra ix. Aragevi.{Hni piBeHb :;:*:iiti:i;ii

6.38. Il.ror4zuu o

B fleperi,rHarorbcfl no npnrraifi

Aea,BeB,-4.8=5cru.

t, AC = 4^tE cv.; BD I flpone4enno CK L I i KB ll I.

AC L

CD =

3 cw.

Toai /ACK = 30' ftyr ABorpauuoro - o.irisiftnraft ryra uinc n.nouqr.rHaMrr i g. KCDB * [p,fiMor{ylHrrn (o6fpynrynaNus Ar.B. y N! 6.37): KB = CD = 3 cw. I L (AKC), 6o I L AC i I I CK. KB I CD, rotty KB t (ACK), onne, KB L AK.

;i{::iiqi:il{ |EOMETPIC SincHiHa O. f,. ra ix. Araseuivnufi pieenu

i{!:

697

3 t AKB: AK = 4 at (erurercrxlrft tpuxyrnur). 3 aACK 3a reopeMoro xocr.trycie: AK2 --ACz + KC2 2AC'KC cos 30'. Ilosna.rrnuru 4na eplnrnocri ei4pisoH KC auissorc,, orpllMaeMo pisasHua:

6.46. B.

x' - 2. 4$. r. @, 2' 16 = 48 * x2 - l2xi x2 - L2x + 32 = 0; xt

6.47. A.

+' = (+Ji)'

= 4,

xz-

6.49.

i-n.noulra-

Horo (x. flo3HaqlMo AJs apyvuocri cropoHy surca suissoro a.

'foti s tACD B.BCD

ACz = AD2

+ CD2:

-B,4 Ha

h:^[ltnr. , ... .. .

{i,fiiEi:iliif: /: i,l\rf .....!......... I : ;

/, ii:41:ii.'..-11!.. ii ii,ili::*:::::::*::

'/l/_ v?:rir!.'-r.....r.'... i.t $a-'....-..'-'-.. . ?i ir;7i1:Jljj.'Ji. ,i rrlf ri'...-..r.r....... ,:,?_{t:..i-.rtr....'... _

ri irt4]'.-.-|:.^...-.-..

= 2cD'zi CD = +

nrolluny -o.

c*r;KM=t2c.].. =!o"-12 2 KN = irivt, + MN" =,ilr, -rX' = jiT+s = tzjz Bi0noeida: l.

6.54. B.

npoercqis noxu.noi

6.59.

6.56. A.

Il.nouluua (BOC) npoxo4rzrr, vepea npauy BO, nepnenn,rollraui ct, roMy (BOC) L s, a BHaqnrL, (BOC) L AO (AO c a). To.qi AO - nepneu4rr(yrflp, noxa,ri, npone4eai B rotlnu -4 go n.noa AB i AC rqnuu (BOC), a OB i OC - ix npoexqii Ocxi.nsxn ,4-B = AC sa yMoBoro, ro OB = OC, 6o pinsi noxu,li uarctr pinHi rpoeHqii. Ile osuavae, qo aBOC npauoxyruufi i pinno6egpenufri IBCO - 45" nyr Haxr,r,ry eigpiaxa BC go nlorquHu ct. a BOC = t BOA sa ABoMa rcareraMu, azrpcu Br4 rrJrrBae pinnicrp niApisrin BC i BA,

6.57. B, B, E.

,qurcyJrspHy

Bidnoaida: o:45".

Toqr{u A 4o n.nou4rznrz cx. Bigcranr, uir+c napa.ne.nrur.rMu rrpsMoro i n.nolluuoro a6o 4eoma [JrorqrrHaMu - AoBxcrzna ix cni.nr,soro neptreEAuKylaspa. trepneH,Ithfiy,'IfiBi4crars uix muuo6i**rr{Mr4 flpsMilMff - AoB}KhHa ix cni.nrHoro pa ni4pisna s xinqauu Ha qux trpsMl4x, nepneHAr4xyJrffpHoro Ao xolxHoi a sux. B

6.42.

6,44, BC L CD sa yMoBoro, MC

6.43-

CD 3a reopeMoro fipo rprr flepneHgrxyrspu, roMy

rxyrcana nigcraHr.

i:i:j;

9 cu; oN -

oR =;BC

raC=s

MK = ,tmoi\o( =,llr"*r' = Jn4+n = Jz% = 15 (cu).

reOuerplg EincHixa O. 9. ra ix. Axa4enirHui

Jl69

= 18

c.v..

K 10 cn

(cnn).

Bi0noeida: A,

6.61. Bi,qpiaox KM ue nepernnae rirorluHy o. flpone_ pemo KK, L ct i MM I cr. fllonllrna, fiKa fipoxoAr.rrb r vepea npami I{l(ti MML(KKrll MM), uicrrarr ni4pi_ sox KM,6o ro.mz K i M sanex<au r1ifi n,rorqr,rni, i neperznae rrJrorqrrHy cr no npalrift KrMr. Hexafi F cepelr,rHa KM, npore4ervro _f'f, t u. To4i -F, e-KrMri KKr, FFr MM1- oi4noni4no ni4_ crani nig rovox K, F i M po nroqrrHr,r oc. (Da.neca

Bidnoai1u: B.

698

6.60- .(o4arroea no6ygona: OK L AC, ON L BC, ropi MK LAC, MN 1BC GTN).

Bi4crans uint 4eorvra roqr{aMu A i B AoB}nI.rHa si,4pisrca.4-8. Bi4cranr nig rovrv A 4o npa*roi I - goexcuua nepneHAraxyrspa, onyrqenoro s roqxu A Ao np,nmoi l. Bigcranr, eig rovxu -4 Ao nJ'rorrlaHr,I o AoBt{uHa nepneHArtnyJ'rspa, npoBeAeHoro

6.58. A,

MN=,lMO,'+ON'=rllr'E

S6.2. BincraHi y npocropi

6.41 . B.

(cu)

6.55. B.

ZBAO = 45" - xyr uixt noxuroro BA i nnorqlrHoro o. BA = a. AC c a, IOAC = 45", AC = a.

ro6ro

6.sg. MN

,12

[oxr.rra, AO

BiOnoeilu:

o=,o-L=Q. turo=nu' cosZBCD=CD = BCJzJz2 cx,

6.52. MN ll AC, rouy MN L BC; KM L (ABC), orx<e, KM L /tlN. MN - ruylcaua si4cranr.

rpfiKyr-

AHanoriqso Mo)fiHa AoBecrr, r4o IBAD - 45'. 6,4O, BO L

6.50.

pirnocroponnifi rpzx;"ruux. AC cct, 6.39, Hexafi-ABC cropiH -4-B B e cx. BD I o,- rogi 1, i CD - npoeHqii i ZBCD xyru, i CB aa rrrorr{rHy o. ToMy ZBAD

yrropeHi ni4noni4iro cropoHaMz A-B

6.51.

6.48. B, A.

BD -- CK = 4 cn a6o BD = CK = 8 cv. BiOnoai?a:4 cu a6o 8 cru.

i1:r,lg::::i::::f:

6.4s. A.

?.".:::p:*". K$MM1. KK. + MMtIt=

--T-,

F, -

aei4xz

cepe4una

K,M,. Toli FF, -

MMr=2ppr-KKt=2.4-l,g

cepeAHfi

linir rpaneqii

=g_ 1,g=6,2(cu).

Bi|noei\a: 6,2 cm. piBeHb

iiiillliii;ill:

:i:iitiiiti

TEOMETPtq Eingxina O.

f . ra ix. Axa4eui,rxr,rn pieexu i{:;: 699

6.66. AABC

ZC = 90", AB = !2 cwt. S e (ABC\. - rp,nMonyrur.ria, nigcranr, Ilpooe4emo SO L (ABC) rovrra S go naorqrznn (/BC). - nigcrani ri4oi4rorxz SA = SB = SC = 10 cn S go nepurr.tr a-A-BC, Togi, OA = OB = OC ax -npoexqii pirHux noxra.nnx na rrJ'rotr{t4rry (ABC}. Togi O qenrp ftoJra, o[rrcanono HaBr(oJIo \ABC, cepeAr,rHa rinorenyara AB.

KRaApar, SB -t (/3C), SB = 7 clt' Ocrci;rutu SB I (/-BC)' to SB -L BD, orxce, nigcranr ei4 rourcu S ao np,nuoi tsD 4opinuroe 7 cn. ,{iaroira.ni rcnaApata neperrrHarorbcs nig lpauuu xyrou: BO I /C. Toai SO L AC 3a reopeMoro npo rpr'I rigctanr ni,q touxu S 4o neprreH4lrnyJTspn, SO ,qiaronari AC,

6,62. A-BCD

BD=ADJ'=8J' no so

=!no

(ax 4iarona;rr rcna,4Pata).

= +Jz cM.

r--- . --. *(+Ji)'

cwr

OB==AB=12:2=6 (cw). 2 3 a SOB: BS2 = OS2 't OB2; 6g=Jrs'-oB2 = J10,-6, =fi00-86="/6a=a

J7'

so =

J4s.n=

=,[sn\ no'; Jsr

{c'a).

Bi1noai0u:7crvri9cm. 6.63. Ilponegemo MO L (ABC) | OK L CD.'fogi MK L CD sa reopeMoro npo rpll neprrerr.414r{yJr,'IplI.

MK * siAcranr ni4 rourra M

4o ctopoura HBaApara' Ocri.rrrcu pinni noxra.ni, nporegeni a o4rdei rovnu, rvra-

rorr pinni npoerqii, a

qii

3a yMoBoro

nigcrani ni4 rouxra

3 yMOK: MKz = MOz

Mo =.lEtr' -

ffiffi* f;$ffif,ffi

a..43C

OK2;

(cu).

6168. V rcragpari MNPK O

ei'{cora

lC=90',

Bepruuu r(BaApara. Bonr,r

4opinnroe no.nosani rinorenysra: t aF = lAB = 18 cu.

Ocxirrru KC I (ABC\, CF LAB, ro KF LAB nigcranr nig rovxra K Ao rinorenyor KF

3 aKCF: KF2 = KC'+ CFz; KF

6.65. Bi,qcranr

rpo rpr4 neprerrArrryJrspi{'

48.

=.EE'-te' =^li?6-3U =.i600 = 30

(crvr).

c*r..

niA tovrcv

Ao ecix Beprrrr{rr :crra/'palaABCD oAHarcoBa

(M q (ABC))'

MA=MB=MC=MD=5cw.

Ilponegemo MO L (ABC), ro,qi O - 4elrrp r{orra, orucaHoro EaBrdoJro ntagpara ABCD, 6o OA = OB = OC = OD

ar npoenqii pinuux V a MOC:

OC =

-2

noxu.nax.

AC =

Bilnoeida:2,5n8

7OO

i:::::

2=6

sxrawr ai4-

41't

=^lno'+oNz

Bi0noei1v: 4

= rl(zft)"

@nrt.

*(zJi)'= G+8

= Jro

(cu).

c*r.,

6.69, .(aao IABC, AC = L0

crra, AB = 17 crrn, BC = 21 cu. V rpzxyrnraxy HaBnporrr 6i.nr,nroi cropoHr.r ,rer$rrrb 6irrl tnuir xyr. Onne, ZA 6i.nrruvft. DA L (ABC). DA = 15 cu. flpone4emo AK I BC - Brrcory rABC, roni aa reopeMoro upo rpra neplerrArxyJrffpu DK L BC. DK - ei4craHr rig rovrcu D 4o cloponra BC. Br,rana.rzmo AoBlrcr,rny DI{ o npauoxyrHoro rprrxytlelzxa ADK (DA L (ABC) , rowy DA L AK) . !,na

naft-

qboro EorrepeAnro euaftgemo :4J(. s OopMyrr.r Ars NJIOII{I TPIIKYTHfiKA:

cwr.

+ OC2; MO = .{Me - OC, i - z,s" = Jr,l '? 5 = z, rJF (cm).

MC2 = MO2

2,5

pi*ni ar noxu.ni,

r,-_.-3a reopeMoro

roqr(a neperzny 4iarona-

KN = KpJ, = 4Ji @yr). on = lm,r = 2r!2 2 3 IAON:ANz=AO2 *ONz;

rprrxyruflrty

/BC

nonigarotr pierii npoexqii. KO = MO = NO = PO sx noJroBr,rHr{ pierrax giarona.tefi rna4para.

ue4iana pinuo-

B npsMor(yrnoMy

nlSA,

-E'(MNP\, nO = 2',12 crvr, MN = 4 cn. AK.= AM = Al{ -- AP - nigcrani ni4 rovxu A 4o nefr. AO

aABC, txa

Bilnoei}a: 30

- Onr; Zs, = il65+r,11(65 - r51 = Jt =

J6,E' Bidnoaido: 6 cttr.

or" - Jfu" - 5' = JL6g - % = nEt'z-' = n

AC = BC, A-El = 36 cm. CK I (ABCI, CK -- 24 cn. Ilponegeno CF L AB, CF 6eApenoro

so =

f"*1.

piruo6egpenzft upaiuorytnsfi.:

cu.

B aSOB: SB2 = SO, + OB2; SO

Bi1noeida: 12 ctt.

6.64.

xtrJurx Ha nrorqr,ruy (zlBC). qenrp KoJra, ofificanoro HaBHoro aABC, To,4i O O cepe4r,raa rinorenyor,r AB. AB = 5 cu (a.A-BC cnrnercrxr,rft), 2

rraBRoJro KBai4paTa ICoJra.

=r=+={=r 22

6.67. IABC 90', AC = 3 cn, BC = 4 cm. - rpaMonyrrrqit, lC =nigcranr S e (ABC). Hexafi SO L (ABC) ni4 rovnra S 4o nroiqtrur.r (ABC). Ocxinsrcr S,4 = SB : SC = 6,5 cu Ba yMoBoro, ro OA = OB = OC sx upoerqii piarux no-

oa=L*en=2.b

nomnoi croponnr r{Bagpara pinni, r:o i ix npoexrleurp Brrucauoro fi onr.rcasoro pinni, ro6to O

,4o

AD=7ocrvr, OK

lcru).

BidnoeiOa: 8 clu.

cM.

rEOMETPI9 EincHiHa O. 9. ra ix' AxagewrivHrf piBeHb:i:::i':::i:i:C:

S^"".

1 AK CD: AK =:2cB

flrorqy

2S*.

Br,r3HaquMo aa Qoprry,rorc

fepona: p

LO+L7 +21 _-rL,

--'

:iiil*it:ii*i TEOMETPI9 EinqHixa O. 9. ra ix. Axa.qeuivmnfi pieenu

l:ifi zll

s*" =

4.3.7

$+14

-s = J+s.r 2.'?

MD = ^tmc"

IcareroM OM

84 = .tEs8 =17 (mr).

o '2=lr =

$.'r.1L1Lr,

rpHrcyrHuHa:

O neperrruy giarona.nefi npflMoxyrHlrxa ABCD nponegeuo treprrenAuxy.nap, Biar,lremo Ha HboMy 4osiar,ny lowy M i s'e4Haervro ii s repurHau[ npeMoKyTHrKa.

S*"

BK; BK = 1o" 2

=2::6 11

2S*" AA

-\44 - rfal= s

(cu).

neptrenAur(yrJrpu BC L c, a aa o.nacrunicrro nepneuArny;r.ffpHilx nJrorqrn (S5.4) BC I g' Ocxialxtt AD L cr, ro ga e,Tacrusictro nepnen4unyaapHocri npauoi i rrrlorqr{Hrr (55.2) AD 1l BC, a a+taqurs, AD i BC .nemarr n o,4Hiit nJrorqrrHl trpsMoxyrHun. "logi ABCD

BC2 =AB2 Bi}noei1u: 29 cw.

i AD2; AC = JrO\rt'

6.72. fleprenguxyrnpui trroilIlrHr o i

= fqll+44L = J841 = 29

(cm).

B nepetttuarorr,cs tro np.sMifi c,

c C

MC2=MB2*BC2=MB2*MD';

7O2

l:::::

AaJIeHa B1A BeplfirlH trpsMOKyTHlria.

i II, (nia.ctaHi ni4 rovox ,4o nrou{r4Hr4 cx). Ocxi,rrxu KKr L a, MM1 L cr i FFr -L u, ro KK,ll MM, I FFr. Ilpoee4euo uepea npaui KKri MMr nrorquxy, eona rreperr.rHae trroqzHy o no rpauiii K,M,. [ei rouxu npauoi KM Hare]rcarb (K$M), rot'ty KM e (K$M\. F, ua.neuczrt nrol4lnai (K.KM) i o, romy Fr-F KrMr. Tanuu uuHom, rpavri KM, KrMr KKr FFr

MM, temarr e o4nifi n,roruuui. 3a reopemoro @a.neca I, cepeArrna KrMr. To4i K,K MM rr'2

FF,

KK'

t=M

*' . lt\o

FF,

cepeAus

:rinis rpanerlii

i;i rpe6a 6y,ro .4onecrz.

6.76. Hexaii rovxa S, trIo Jaerfirrrb rrooa trJorr{[Horo trpflMo]ryrHoro rpraxyruuNa /BC (lC = 9O'), BHaxoArlrbcs na nigcraui 10 cu ei4 nontnoi fioto eepruzuu:

SA=SB=SC=10cm.

Ilpone4e*ro SO L (ABC), rogi AO = BO = CO nn npoenqii pinnux noxzJrr{x, ro6ro ro.rxa O qeHrp onucaHoro HaBrtoro rpzxyrHzxa-43C ro.na, V npamoxyrHouy rpr{nyrHr.rxy eiH ler*curs ua cepe4nui rinorenyaa.

M@s,MeP.

Ilpose4emo MB L o., MD L 9, MC L c. BC .'._ npoexqis MC nannonluny ot,, roMy 3a reopeMolo rrpo rprr neprreHAnftyJrfipu BC L c, a sa s.nacrunicrlo trepner{Aur{yrrflpHux rtJlorrlnu (S5.4) BC I o. Octti:rcxn MD L B, ro ea e:racrznicrro nepnenguxylapnocri npamoii i.n.norquHr.r (55.2) MD ll BC, a asa.rrzrr,, MD i BC rexarb e oAHifi naouzui. flpflMonyr]rux. Toai MBCD

Ocmirsxn aiarosaai npfiMonyrHuKa pisui i rovnoro neperuHy 4irRrscn naenir, AO = BO - CO = DO. "Iogi rAOM = rBOM = tCOM = aDOM Ba ABoMa KareraMr4 (rpzxyruznu npmronyrni, 6o MO L (ABC), or*.e, MO I AC, MO L BD), 3 pinnocri rpuryruzxin nunnwnae:. AM - BM = CM = DM, ro6ro roqrca M pinuonig-

6.75. ,{aHo ni4picox KM, au:aft He neperuHae nJaon{r4Hy cx,. qepes Hinqi K i M ra cepe4zHy F ni4piana I{M : trpoBeJrr.r nepfienAraxy!'rsp[ Ha trroqqxy a: KKp MMr

12 (cu).

npo rprr

6,74.3 roqnz

6,71. Il.norqunu o i 0 neptreHAuxyaspni i neperunarorsca no npauifi c. A e a, A e F. Ilponegemo AD L a, AB L9, AC L c' npoexqis AC Ha n,rouluny B, touy 3a reopeMor) BC

-AB2.t

rorrfta neperrHy

rreHHsM trepleHA[KyJaEpHrrx nJlorqrrH.

Bidnoaida: 9 an.

AC2

35 (cu).

MOLAOiMOIDO.

= ^tEr -1L

3 t MBK MK2 = MB2 r BKzi MB = ^[MK" - BL: = jrb'z -1r'z = tEE

Ilrorqzna (.48C) neperzHae nr'roqr4 Hr.r (AMC) i (BDM) no rpnMux AO i DO. AO L DO arc npsui, rqo uicrarr 4iarouali ftBa"qpara. Togi (AMC) L (BDM) Ba oBHa-

3a OopMyroro lepoua.

2.Lr-B = 66 (cu').

flronlraHrz (AMC) i (BDM) neperwHar)rbcn tro npamifi MO, MO L (ABC\ sa no6ygonoro, rogi

22. s*o" = @

Jts? -:.:1rG: -14 = Ji{+g = s

rinorenyaoro.

4iaroHa"re* HBa4para.

Ba reopeMoro

+ rB + 20

Toti AO = BO = CO = DO. O

ei4 rovnu M napu MK I AC, MK - ei,4craHr go uaiimennoi cropoHu AC. ni4craun aig 3a yuoeoro MK = 15 cv'. MB ro'rxu M Ao nrrorqrrHtr (ABC). Suaftaevo MB B npsMoltyrHoro rputtyruutta MBK.

fkorqy aABC snaitgeuo

- Ilo6y4yemo MO )- (ABCJ. MC = MD. IAOM = aBOM = aCOM = aDOM sacninr;eumu

MB L (ABC\. Ilpoee4euo BK L AC - Br{cory trpo rprr nepneuAur{y-

B (DopMy,au n.irorqi

,Lr,

BC = 20 cu, AC = 11 cm. lB - lr.afi'mesllnufi, ocxi.nr,xu rerrtI'rrb HaBnpor[ uesmor croponra AC.

BK anaft4erro

lT?,

nBarpar, M e (ABC), MA = MB =

6.73. ABCD

6.70. y aABC AB = 13 cm,

*ABC, togr

Bidnoeida: 35 cv.

= 84 (cn).

dx =2 8i = g t"*). 2t 3 aADK DK2 AD2 - AKr: - # ='lrrb. DK = ^[AD\ AK' = "{F Bi0noai}u: \7 cw.

'Ma,

;EOMETPIS Einnxixa O. 9. ra ix. Axa4ervrivHl4i piBeHb ii:ii::ii:i:i:::i:i i

OB=1AB =6 2

S aSOB: SO

cm.

SBr=SO' + OB2; SO= JSB, AB' =t0' -6' =8 llorqzHu (-4BC).

(cm).

mynaua sigcrans si4 rovxrz S 4o

ABCD

pom6. M 4 (ABC).Ilponesemo MO L (ABC) i MK L CD. MK BiAcraHr nig roqxv M Ao cropoHr CD. MK = 8 cm. OK L CD Ba reopeMoro npo rpz

6.77. n

epneHAr4KyJspu.

:iiii;:ll{l:ll TEOMETPIC Eingnisa O. 9. ra ix- Ara4ervrirHr4fi

piBeHb

iii{i

7Og

Bi,4cranr OM s\F,ro,txu M uoxna snafirz 3 np.fiMoxyrHoro rplrrytuuxa MOK' Ilonepe4nro exaftgeuo AoBrnl4Hy ni4piaxa oK. lde uoxna apo6urv, BI,IROprcroByrorru nnouly a COD. ,{iarora.ni pou6a nepneugrrcylapni, ro'{rca O gi.nunr ix naenir'

tcoD: oc =f,ac=

oo--!no-r]=a

MN LAD,- MN BK, MN = BK, pe BK - nzcora rpaneqii. lryxana eigctaHr ui4 touxu S go cropoxu rpaneqii, 6o SM L AD sa reoSM

peMorc npo Tpr{ ilepfieHArrrcynnpu. ocxi.nr,ria n rpaneqiro cyMa AoB)r(uE ii ocnos gopinnroe cy*ri 6iunux ctopin:

-T

c*r.

AB.sin 6o' = n

= Vi1?

-8)O?+8) =,lg.r5= 3.5

(cM).

=\Nttt 2A

= 3175

SOK SK2 = SO2 -f OK2;

Bi)noaidu:

= Jca =

(cu).

cm.

cm,.

A-BC.

"^=#,

civr.

AB2 = az =

fr. =#=

r08,

,48 = u1* = 6Jd

(crvr).

Cxopncraemocs tpopuynoro ca,rencrrocri cropoHrr rrpaBHJrhHoro rpuKyrHr.rfta i pa4iyca -.t

onr{caHoro KoJra

-,[N

- Ao'

Bidnoaida: 8 cyr.

AB ._=o 6vE RJ5 ,ars sHaxoAXceHHs oA: oA =-= V3

SO2;

AMr@BC),AM=5cm. i

ninnepu-

Merp rpnl{yTHffKa.

iii: fEoMETPlg SinqHixa o. 9. ra ix' Axa4ervrirHufi piBeHb ii::.:.r:{;ii:ii

rue4iaua

M 5cm

- Br4 cora, a Bnaqrrrb, A IABC, CD = DB = I BC = 4JT cv.

flpone4eno AD

KoJIa,

(cM).

t0' -6, = n{00 -86 = 162 = a 1cu;.

6,42. ABC - nparr-rmHuft rpr4nyrnrx: AB = BC = ,4,C = 8Jd crvr

ML= MF = 17 cu. ToaiOK -- OL= OF, rouxa pa,4iyc r{oJra, Brrr,rcaHoro n rpr'rxyrnux -ABC. O ,[onnruny ri4pisrca MO zuaftgewo s tMOL' pagiyc Brucanoro . = { , g" r p

3 aAOS: ASz =AO'z*

y IABC AB = 26 em, BC = 28 cw, AC = 2O ctt. M e (ABC).Ilpone4emo MO I (ABC), MO - niacraHr nig toqnra O Ao flJrorqurrt4 Tprtryrnr{na. flo6y4yeuo MK L AB, ML L BC, MF I AC, MK, ML, MF ni4ctani oig ro.rrcra M 4o ctopin tpurcyrnuna. OK, OL, OF * ix npoerqii. 3a yuonoro

(crvr).

8nafi4euo cropony rpr4nyrnr,rra a @opmy,ru fioro n;rorqi:

6,79.

snafigemo s (popMyrl4

so= Jsr, -oK, =Jro, -6, =Jr00-g6

saelcoao

6.,6 (cu)'

Bi4noeidu: 6 cra.

36J3

Hexafi SO L (ABC), rosi OA, OB i OC piniri npo'ercqii pinulrx noxr,rJrux, O - qerrrp KoJra, on[cauoro

Sttt=.ISO'+OMz = ya'z*(a./a) --Jg+zt =J36=6

,l1ot--Jt

AS=BS=CS=10mr.

+ OM2i |

7o4

^m" -8"

,lllt" - Ot';

pinuocroponuifi, Smc = 27JB 6.81 . ;ABC Toqxa S piuHori44a.neira nig nepurran a-43C.

24 (cwl,

AB=L.24=12

NM = BK = 6V'B cru; OU

Mo]r{Ha Brrr4caTI,I I{OJIO,

AB + CD = BC + A-D =

SO2

(cr,r,).

(cm).

oK =21n.*:= P*""

3 aSOM: SM2 =

sr6 = 886

P*"" = 36vE cm. S e (ABC). flpore4euo SO L\ABC\, - pinnocropouNlit, l/3, SK I BC, SM IAC. To.4i SO - ni4cranr ri4 rovrcr S no nnon\uau (ABC). SN, S,M i 51{ * nigcrairi ni4 rourcu S 4o cropin IABC. ON, OM i OK npoex- ix piniriax qii. Ba ynnororo SN = SM = SK = 10 cn. ToAi ON = OM = OK src npoexqii noxr4Jrux. O - qeurp Bnrrcanoro s aABC no,na, ON, OK i OM - fioro paAiyctr. AB=BC=46=Pwc -=36'F g =12V3 (cu). B t; '*Z*" L44.8 = 108J8 (cur). ^ = AB"J8 S*. A

OKz;

pinno6iuua rpaneqia. AB = CD. P *"o = 48 cw' 1A = 6O"' 6.78.. ABCD To.rra S He Jre)fir4r6 y n;rorqrzni rpaneqii. flporegemo so I (ABC). OCni.nrHra toqxa S piBnosiAAalena aig cropin rpaneqii, ro i tovrca O pisuoniA,qarena ri4 nzx, 6o qeHrp noJra, Bnr{caHoro n rpaneqiro. pioni noxu.ni uarort pinni upoerqii. Tomy o cepeAfiHa ni,qpioxa MN, qo e'eAnye cepeArrHr'I ocnon rpaneqii' To.*ca O

#1,

= Co, 16 = 6,4 BiinoeiOv: 6,4 cw.

,ABK: BK

6.8O. tABC

(cm).

'Jlisk.:t !t.c.r.:.

tE.r6:r+n

Bidnoeida: 15 cn.

Mo=.tMK'-oK' = l7 4,8'= Ji8-4fX8+4s) = JtTLr,s

= 42 (cn\;

6.8=24

3 tMOK: MK2 = MO2

,1r r-nski-r^.rq.ir-.-..

3 a MOL: ML2 = MO2 + O.L2; MO =

(crvr'z). A s*- = tco. oK; aK ='* =# =4,8 (cu)'

l1;,Ii.lr,i;o.':.:.i:r.:,:

26 + 29 + 30

ot=Ssa =8 42

1"*1;

<"*t.

oo=t

s*" = @

cD = Joc' * oD' = J8' + 6' = Jroo =.ro (cm).

s^"oo--!oc

fepona: o

Ba rpopny;roro

BC, rogi AD

3a reopemoro npo rpr4 nepnen,qrrxyJr.fipu MD L BC, MD eigcraar rig rovnlr M 4o croponu BC,

:itfilfiili

TEOMETP|9 Einsnina O. 9. ra in. Axaseruivnrfi pieenu

23 y"i rAg lo M., q.

{!}

7Oa

6.98.

3 tACD: r-L

AD =

-=-=,-

t@ ID, = Jta.F)'-(+S)' = Jta,n aG)(sJs

,FJi tif,s

-.'

aJs)= S^*"

(cM).

Morxsa taxo:rc suafirvt.A'D inanme.

-C

60', AD = AC

3 tMDA: MD2

sin

t; 60' aJs Y2

A) = 12 (cu).

AM2 + AD2; MD = E

+lr'

6.97. A.

ABCD

swr

BM L (ABC). MD = 39 ctt. MD

- rr5

12 cu

S.or" = S"o"" cos o;

= Jr296 = 36 (cm). Bidnoeidu: 36 cwr.

"oro=ffi

npauoi a.

KM - ei4cranr ni4 rovrcu I(

* OM2; oK - ^[KM' - oM' = JB" - 5' = ^fi 6, - %

='1L44 = t2 (rltt\.

Ridnocida: 12 ctt.

6,85. flprui m i n unno6ixtni; n c a, m Lo', A € m.

roqxa neperrrny npamoi m e nnorqlrnoro tx. To4i ei4 rovnrz -4 4o n.norquurz o. OA = 6 clt. AO ei4craur mirx mumo6ixnrauu Ilpone4emo OK L n, OK trpsMuMrt m i n (ix cni.nsszfi nepneH4rzxylcp).

AI{ L n, A-K 4o lpamoi n, AK = 1O cxr.

3a reopemoro npo rpu neptreHArrnyr.fipu

BiA-

craHr eig ro.{xz -4 3 yAKO: AI{2 = AO2

r OK2; oK =,tqx' - do" = v[0' -d - .,[00 -36 =.,62 = s

S.-.

= S"no, . cos

"uo"

An. nc

,!i

6.101. S€'*

45';

DC, rouy KD

{--

(crur,):

6Oo

DC (aa TTII).

-L

"flTr".

=V=*f

= Sor"r'cos60o. 1

S*,,, =180 t=90

( \

=B2li

@nt).

S*"o- AB

CK = 15. 12

180 (cu'?).

(cv'?;.

Bidnoeida: B.

6,102. aABC i IA.BD pirnocroponni. AB ' AD - BC l',6 cm.

1cu).

Bidnoaida: 8 cn.

flpoee4euo CK L A-8, T oai DK L AB ea reopeMoro npo rprr

6.3. OproroHarbHe npoe(qitoBaHHe onpeuufi BfltraAofi napa,le.nsuoi npoexqii, noJru Eanpau npoertqiroBaHH,tr trepfieHAuxy napuuit Ao nroilIuHtr npoeHqii' Teopeua. Illorqa oproroualrHoi npoexqii ,qoninsuoro MHOTOHyTHIIICa Ha ITJIOIqIIHy trJIoIqIIHOIO 4Opinnroe 4o6yrxy n,rorqi cauoro MHoronyTHuKa Ha ROCIIHyC Hyra uirx @irypn i n.nourunoo nPoexuii'

Opmoaouanana npoettutin

S'=S.costp.

6.87. B.

r2Js

Bi}noai0u: B.

ni4crans

Hexafi O

6.86.

' 1t t lll cosa = =

6.1OO. .41{ L (ABC), AD

3 aOKM: KM2 = OKz

= #h Tr, + t:,"

183

6,84. flpaui aib - unuo6ir+rni. aca, b ea,b La- K e b. n n,lorquui cl OM I a. "logi KM I a aa reopeuoro Ilpone4euo rrpo rpn rreprelr,quxyrrspu.

s-r" = &le

84 _: = -10 5 B)' cosa = I = 1, 15 3 93 R) cosn=l=:. '21 7 _61 ll '122' cos0= = :

.,T5rl

cos cr;

A) cos0 =

noxlrJra'

cwr,

(ABC).

3 a MDB: MD2 = BM2 i BD2i MB = "tfio" an" = ^tEs" ,%

(cvzr: - cos4D"-18;2,r2y[2 \12 18 = ra:1 = 36 (cu2). B) Swr =

npoen4ia uan;totqwuY BD -1i 3 "ABD: BD2 = AB'? AD"' + 144 + 81 225.

fifi,ffi

cos

trpsMoxyrgr'rn, AD = BC = 12

- S^*.

AB--CD=9cw.

BDz = 122

E) s,,,*

,lrb + 144: .,[ 69 = 18 (cu).

BilnoaiOu: L3 cv.

6.83.

6.89. r.

6.91. B,

6.90. F.

6.92. A.

tr.

rprHyruzxin

PosrrsneMo rBCK. KC

ICKD = 6O" Hyr

nepflneH,4rrxyJT,flpu.

nJrorIZHaMu

-4-BC

mi:H

i ABD.

Bucora, a BHaqurb,

ue4iaua tABC; IKBC = -6O'.

KC =

-f3

BC.sin60'=+Ja *=

,ABC =

(cu).

aABD aau*o"on1, uo = KC = 6

cvr.

3 tCKD aa reopeMoro xocuuycir: CD' = CK' + DK2 6.93.

6.94.

.U.

7OO itli reOnnefplg Einqxixa O. 9' ra in. Axa,qevrivHnfr piBeHb:iii:*i:::i:i:il

- 2CK DK.cos

60o;

CD2=62+6"-2 6 6 1=36+36-36=36 CD=6cw(Cr>0).

6.95. B.

BiOnoeiOa: 6 cm.

iiiiiiill::li 23*

TEOMETPI9 EincxiHa O. 9. ra ix. Axa4euivxnfr pieenu

ii;i:

7O7

6.103. aIB,Ae cx,,Be B.

exqis.

ToAi

s*",

S4e,c,D,

Bi}noeida: 18 cpr. 6.

ftoro oproronaJrbua rlpo'

S-",'cos3o'.

6.8.sin60" = u,

t; =

24JB 'rcvt'). saa",r, = 24.6

ao <.*'1.

Bidnoai1u:36 cvt2. 6.1 O5, Hexaft ABCD

gaua rpaneqia , A\BrCrDl

ss(,, cose. "o",1 7+9 a+b h=;.5=40 S*ro=if

sr,u,.., =

!rre' * acD

oproroua,'ii'na npoerqis'

cosg=

20 40

nipaui4r.r pieni

09,

Poarr nHeMo AJts cpylnocri oproroHa,rb-

Hy npoexqiro 4aroi rpaneqii ABCD ua naorr$4rry o, gxa micrurr ocnorylS i napa;re.nrHa ocnosi CD. Ilporegeno DD, L a i CC\ L a. ABCTD, oproroHaJrbua npoexqia rpane4ii ABCD ua nrorllrHy d. Ocni.rrxr,r DC ll a, ro DC ll DtCr DIDCC. rrpffMonyrnrrx, Dpl= DC = 3 c1a. Ilpone4errno

= 60".

D,E = ,[AD: * AE' =

niparui4ra SABCD, Hexait SO - arrcora SM IAB, SN I BC, SP LCD, SK LAD. ToaiOM LAB,ON LBC,OP LCD,OK LAD

B a DED,:

3a TeopeMoro npo TpIr fiepneHA]',rrcyJrspu.

ISMO, ISNO, ISPO i ISKO - ABorPaHHi 3a yMoBoIo nonu pinni. Kyrll nptr ocnoei.

Tod aSMO = aSNO = aSPO = a SKO

i uporu.ne*car{M roc'lrpr'rM nyroM. - cni;rrnxfi) Snaqrrr, oM = oN = oP = oK, ro6ro 'ro.rxa o pinnoaig4a;rena ni4 crnpirr tlorupuBa r{areroM (SO

xyrwxa ABCD. O - rlenrp KoJra, Btrr{cauoro 6,1O7. Ba yrroBoro (/3S) L (ABC\. SO sncora

s ABCD.

c (ABS).

KyrI.fipu.

cni.xlnr'rft)

nporunerxuram

.o". onn, = lE 3"8 "6 IDED,=t rr""o. Jt DE= i=:. 4

CMB lrpw napaJrerrbHr4x npsMrzx PK i AB i ci,a:r.ifr CD. flporegeuo MO L cr. To4i NO npoercrlia nparroi CD Ha nJrorlzny

cx,.

xyr

rrlo Jre)firrrb HaBnporrz Hyra 30o.

tMOC = aKOC Horo r43.

=8.,6 (cu).

,[MF\ Nr' = ,17 + o' = oJz. SaNMo: Mo=Luu=ofr sH HaTeT, 2 2

- Bnigclr OM = OK, xytort.

sa KareroM i rinoreuysoro. 6icetrpuca rcYra C.. To,qi OC to'rxa Taxunr vuuom, ocgoBa Bt corrz O

pinuo6iuna rpa-

6.110. 3a yMoBoroS ll cr. AB nCD = M,CD n o =N. ICMB = 4b". rlepea npaui AB i CD, ulo neperrruarorr,cfl, [poxoArrrb rrj'rorquna (ABC\, ax,a neperrrHae nrrorqr{Hy cr no npnuift PK, PK ll AB. IMNK = 45" strt ni4noeignnfi xyry

ISMO = ISKO - pinni 4aorparHi ny'tu, y"reopeui rpannur 3 ocnoBoro. ISMO = a SKO sa rocrpr{M

= 30' - xyr uirrc npauoro CD i n;roulunoro o, Ilpone4euo MF L PK, OF L PK sa reopeMoro npo rpt4 neprrenAzlcyJrffpu. To4i MFO -;risiftnzji rcyr ABorpaHHoro Kyra, yrBopeuoro firou]rrHaMrr (ABC) i a, flosHaqfiMo MF = a. Ocni.nuru s d MNF IMNF = 45", ZMFN = gO., ro i INMF = 45", ro6ro a MNF pirHo6e4penzfi a ocgoBoro NM: NF = MF = a.

LAC i SK L BC, roaioM LAC BC aa reodeMoro npo rpr{ nepnenAil-

KareroM (SO

J6' -s'= JB6_o

IMNO

flpone4ervro SM

rpo rpu nep-

reopeMoro

IDED.

npsuzuia ,43 i Clrl). 3 qboro Bz rrrraBae, ryo AD, = BC r, ABCTD, neqin. Ocriarxr,r Ba yMoBoro e sei MoxHa Brtucarvt rcoJro, To AB + DtCl = ADt 1- BCr 3 + I = 2ADr,,4), = 6 cm. S rADrE: ADI - A5z * Or[zt

ocnoei

co6oro, ro n ocrioni .neIcI{Tb MHoro}cyrur,rx, y .arcufi Mo}rcHa Bn}rcaru KoJro, a ocnoBoro urcoru nipaui4h e qeHTp qr,oro KoJra. ,{onegelro qe.

rorray SO

o,a

3a e,racrilsicrro pinHo6ivHoi rpaneqii AE = BF = *;o' =T =B (cm). To4i rADrE = dBC.F aa ABoMa KareraMu (DrE - CrF ax ni4crani minc napalemuramlr

rvix

nipaui4u. SO L (ABC\,

DE L AB -- Br{cory rpaneqii

ABCD. DP LAB

mixc n,no- ryr ulrruoro rpafferlii i nrorqznorc lpoexrlii. Ana.noriqno CF LAB, Cl IAB.

Bi4noeida:60".

6.106. frcrilo 4eorpauni xytu npz

rreHArarcyr'rffpfi. Togi

(crvr'?).

B e a, C e AB, AC : BC = 3 : 5. I ll BBr Togi AB, npoexqia

ei4piexa.43 Ha trJrorqury ct". Cre-ABr Ocxi.rrxia CCr ll BBr ro aACC. o al3B,. AC : AB = AC, : ABri 3 : (3 + 5) = AC, : 48:' 3' 48 AC. = = 18 (cut. '8

4anui napa;renorpaM, aAlBlCrDr

cr,

Ilpouegeuo

" CF-\D

rrron{rru (55.4) AC L BC. 3 aABC: AB2 = ACz 't BC2; BC2 -- AB2 - AC2 = 13' - 3' = 169 - 9 = 160. 3 r BCD: BCz = CD2 * BDz; cD = ^[BC' - BD' = Jtoo - 16 = ,1144 = 12 (civr). Bidnoai1a: 12 cv.

6.104. Hexaft ABCD

6.108. A e

fkorqranu o i B neperr,rnarorBc'a rlo npsrrifi c. AC L c, BD L c. AC = 3 ct*r, BD = 4 cll, AB = LB u. 3a sractr4sicrrc nepnengtxy;raptocri

leperrruy 6icexrpucu xyra C ai cropo-

7oa iii: TEOMETPIS EincxiHa O. 9. ra ix. Ara.qeuivHxfi

piBeHb :iilii'.::.fi::r

3 t MOF: sinlMFO =

.- a'r2 MF= 2'o= -'',tE' 2'

ZMFO = 45". II{o fi rpe6a 6y,ro 4oeecrz.

iiili:ltiitil

TEOMETPIC EincHixa O. 9. ra ix. Axa4euivnrafi

piaexu;ii:fi 7Og

6.113. IABC i rABD

AB ocHoBa. Ifpone4euo CK I AB, roxi CK Bhcora i megiaxa pinno6e4penoro IABC, rouy K cepe4uua AB.

Arfi 3p)rllHocrl oprorouaJrbny npoexqiro Aairoi rpaneqii ABCD sa nno' rrluHy ct, ,srta uicturs ocuony /-B i napa,'re.irr-

111

Posr,T,aHeMo

na ocnosi CD.

i CCr L s- ABCTDT npoenqir rpaner{i ABCD sa

flpore4euo DDr L cr oproroHaJrbHa

a, ro DC ll DtCr DtDCCt

^ABD KB - me4iaHa, a sHaqrrrb, i nHcora: KD AB. ,zCKD = 0( - Kyr vix n,rorqznauz

(ABC) i (ABD). flpone4euo CO L(ABD), IAOB oproroHarbna rpoercqis aABC aa ntot4uny ABD.

trp.fiMonyru[n, DrCr= D( = ]' sNt. Br4cory tpaneqii Ilpoae.4evo DE I AB ABCD. DP LAB 3a reopeMoro npo rpr4 rep-

mint n.noneHAnHyJlflpr4. 'IoIl IDEDt - Hyr rrlrHolo rpanerlii i n,ror4r'ruoro npoexuii. Asa.noriqno CF LAB, Cf IA-8.

Ba n.nacrneicrro piaHo6ivHoi

rpaneqii AE - BF

*#=+=B

t: DE.--t.-:

3,13 --

=b 3aDED.: ' cos 3U" J3 t'o ABTDC .DE="'-1 .6=36 -.s --- ='*':- 2 --MtD 2 6.112. IABC i

=.t6' -l'z =-vE6f

sr6

rrtrrHaMv (ABC)

3 IAI(D:

AD2 = AI{2

dx"

=1DS

1cn).

-7

Bi}noai}o: o =

'f KD2i

arccos-

- DCz

352

'aft

y N!N, 6.112, 6.113.

DC' = KD' + KCz - 2KD . KC . cos a;

5' - Bb'

2.35.5

cM,

710 iii: rEOMETPIS Einssixa O. 9. ra ir. Axa4ervlivHrf

piBeHb

:i::::::;i:i:::i:::::

6,1

arccosl:

15. IloscseHHfl

74'

KD =,7AD'-

(crr,r').

L= 3o 74

(cu'z).

cu'2.

AuB. 4o Jr&i\! 6.112, 6.113.

3 IAKD: AD2 = AK2

ax =!ln= 16 2

420

72. 35 =

s-uo = SLrr'cos0 = 42o'

KCr+KD,-CD2 ,2b+35', o=u"""o"l. *'"""" ' - 2EC . KD 14 2.5.35 =!,14 1 (c*t). .cosfl=Go Ao rnv2r s (cv'?). S*or = KC = 12'5 = 60 =sS*, coou - -- -L S*. Z ."' 14=!9 iAB =

cn.

35"

Mararouxa

=Lln=L2 2

+ KC2

,*" =**

KD = "tqn', JrF -rr" : JGr ' rrr(vt +rD = Jr* 4s - 5'7 =35 (cu). 3 aKCD Ba reopeMolo rocunycie: CD2 = KC2 + KDz - 2KC'KD'cos u;

Bi1noai1a:

(cMr).

6:2

2.KD.KC 1 cI' = arccos-.

(cv).

cosc, =

c = 1oo8

cos

Ba reopeMoro HocrzHycie :

KD'

='ll}'-lr'

14. O6fpyrryBaHns

3 t KDC

OcHirtxz CO L (ABD), (KCD\ L (ABD\' CO c (KCD\,roOeKD.

KC=.tAe -AI<"

aTCCOS

=Af{2 + KD,; KD=^tqn, - ax" =lffi, -Ir, =,lr|.49 = 85 (cu). 3 iAI(C:AC2=AK2+ KC2; KC=^ttrC -dX'=1T5'z -n" =5 (cu.).

oproroHaJlbHa npoexqin

ACr=Af{2 + KC2;

AD2

"AOB tABC sa nrourrauY (,4-BD).

-:AB=24:2=12

2.63.12 2.63.12 2r',

B oAKi: ax

nYr uinr n.no-

e KD.

:. - 7 rc*'\. 21 Bidnoeidt,: o ,.""o.2, 5=9I? 2l 7 "*'.

Onyctuuo nepneuAurdy!'r,flp CO na n,roqrzny

3 IAKC: AK

s-o, s*.

(ABD).

s^,,. = 1 AB. KC = 16.68 = tooS 2

6.1

Brrcora flpose4ervro CK L AB, roai CK i ue.qiasa pieso6e4peuoro \ABC, rowty K cepe4nua A,B. V tABD KD megiaua, a auaqurs, i snco-

ra: DK LAB. zCKD = a

COSd

(cru)'

(cM':).

cri,tr,sa ocHoBa.

= ! an = 16 cru.

(cM).

aABD pinHo6e,4peni, ,4-B

L (ABD), (KCD) L (ABD\, CO c (KCD'),

B ;AKC: 'dx

KC=,tAC - AK, =^tab, -16" = Jaf .+O-OS. |AKD:AD" =Al<'+ KD2: KD= JAD,, _ AK, =lTO, _16" = G6.4 = 12 (cu). 3 t KDC:, DC2 = KC2 + KD2 - 2KC . KD . cos a; 63' +lz2 - 682 L44 2 2

(*r). ro4i

ea.4BoMa rcareraMvr (DP = C_r-F ax^niAcrani uirx napa'neJrr'Hr{Mr4 npai.mu AB i C1r1). 3 rlT'oro Brtrrrr4Bae, t4o ADt = BC r ABC rDt - pinno6ivna rpaneqis. Ocni.nsnr4 3a yMogoro e sei moxna Blr4 caru KoJro' To AB + DtCl -- ADt + BCl, 3 + I - 2ADrl,1 = 6 cM'

F4 -;A

Ocxi.nsrcu CO

AC2=AK2 + KC2;

tADrE = dBCrF

3 aAEDr: Mi = Aa" + D,E2i D,E =

IIJIOIIII4HY d.

Ocrcirr,xu DC

pieuo6e4peni,

cnimsa

KD2;

cm.

AK' - Jro' ta' €6 =

= !@o - 16)(10 - 16) = J4 = 2.6 = 12 (crr).

i{li{l:{iii{t rEoMETPle EincHiHa o. e. ra ix. Ara4ervlivxvrfr pieenr {:]dl 711

't AK2

B yAKC: KC2

KC = ^[AC'

3 r KDC

9acrrarla 2

= AC2;

- AK'= {CL 1d = j(65 -

Ba reopeMoro

KD2 + KCz

nocrnycis:

16X65

DC'z = KDz

- DC2= 122 + 632 -632=

z'L2'GB

"o"a= ,.1qp.1{c

+16) = J4r â&#x201A;Ź1 = 63 (cu).

- 2KD ' KC ' cosa; 144 2 o= arccos 2 2l 2.12'f3=^

+ KC2

17. Posn'saanua i4enruune 4o Jrl! 6,69.

18,Il;rorqaga"o*ottp"*o*yrnr{RaS=4.6=24(cvr,).flrorqanpoexqiisr=12"*2.

Js,1211 =

cosQ =

Z+=

arccost = ti0".

s..^- = -'AK.KD= 16.12=192. 2

s*" =s*"

coso(=

rcz

+=X.

s=-384 2t

BiqnosiAb: ct = arccos 21

I a, MA i MB noxra.ni, OA i OB ix rpoercqii. MO = 4 cw, ZMAO = 30", ZMBO = 45", IAOB = 150".

19. MO

Tecrgnn caMoKoHTPonK) r.lacrnxa

1. A.

3. B.

2. [,.

4. B.

6.oB=R=*=LE=Js JsB

(cM).

aMoB: Mo

= ^[MB'

- oB'

Flfl.$H*iii,li B:l?'xlri!.:'.'.'c:

i{ln\S;iiiil'

tlllii,:i:;ii'lii '/;ri$'.,ri.IJ....

Ga+s2r6

J4-3

10.

t---f-t_l t

12. BD = ali:

ta' -Arr = "ltz'; l, Bidrtoaido:

J144 + 256 = J4o0 =

20 (cu).

(cnn).

VTABCZC=60.:2=3O'.

Bidnoaidu: 2O

s!2 2

f^2

3a2

= A-l\t r

(-4-B

cwr.

- xarer nporz xyra

30o).

i B neperuuarorr,c,ff flo npruift c. flpou, BD L P, BAi BD nignoei4no ei4crani ni4 rovrcrz B 4o n.rrorqun cr i B. ffurorquna (ABD) npoxo4urb rrepea npaui 243 i BD, rouy (,43D) L a i (ABD) -L B, ro4i (ABD) L c. p. IACD = 60" - rcyr uirx naorqunamu cr i d BAC = t BDC aa KaaeroM i rinorenyaoro (BC = cni.nsua riuorenyaa, BA = BD sa yMoeoo). = 16 cm ne4emo 8.4

"Jo I

ZBCD

13. B.

=!22

^co=

30.,

BD =

!ac

--8 *r. AB =BD = 8 cv.

Bi0noai}u: 8 cttr.

22. a LB,Ae o, B e

14. Bacora rplrr{yrnt4fta: h = ^,82' - 6' = JT44 - 36= r/i08'

AII 3.F . Srn@,h6J32 =:: =: = 1

AA,

= 30".

AA, = 3

cur;

F=c.

BB, L c, BB, = |

sNr.

=.,,[T

AA, e a, AA,"*. L c, rouy,4-4, J'

Bidnoei0u: B.

orxte, AA,

ArB.

3 aABrB:

15. A.

ArB = B.

712

2 413.4. cos30"

"*.

AC = 2AB = 20 cu

21. fllorqunu

oB, v__,+ BO' = ,la' -_r = .IBB: \ "

16.

V48 + 16 +

2O. fl.norqznr,r o i B neperzxa"rorr,cs no npamiii c. flpone4emo -4D L a, AB L I, AD i AB ni4ironi4no ni4crani ni4 roqnu A 4o nnorquir cr i -B. Il.noiqr.ria (/.BD) npoxo4lrrr, qepeo lpav.i AB i AD, rotly (ABD) I a i (ABD) I B, ro4i (ABD) L c. 1"DCB + 6O' - xyr uirx n.norqruarru cx i B. cni;rldABC = aADC salg"anerorr i rinorenyeoro (-4C na rinorenyaa, AD = AB = 10 cel).

= 1 (cru)'

f A--fl_l t--T-t--l

Bi1noai1t':

2AO . BO. cos 150'

\=[a-aJV

Bidnoai|u: 4J7.

AO'z + BO2

I A-T-tl

ll| 'illliir,l;'i:l.ii lel ,r;ii..:r:J:::::::::lfl6l

BOM: OB = OM = 4 cu (npsuonyrHzft piano-

5. A.

'.n."

r"*r.

6egpenrafi rpuxyrnzx). 3 tAOB 3a reopeMoro rcocnHycin:

Bi1noei1a: A.

7' h,li:n,n.,..,-....-'

Mo lo= te30" =!.3 =126=+,8 JB 3

BaAoM:

i::l: TEOMETPIq EincxiHa O. 9. ra ix. Ara4ervrirHui piBeHb

iill:i{i;ii::i

fi{iii{il:ii

JTq *

BBi =

"41-

=.@

sJ5

(cm).

TEOMETPI9 EincHixa O. 9. ra in. Axa4euivxr,ri piaexu

ii{:;i 719

3 .AA,B Au =

rtr4

- 4n', =,tl',;

n = t6;n

= 16o =

26. ABCD

Bi4noai}u: 6 cv.

23. fl;rorqnnz o i

B nepernrarorbc.a no npnulifi c. A e

Touy

c. aADC sa KareroM

(,4-BD)

tABC -

i rinoreiyaoio

(AC

cni.nrsa rinoreHYea, AB: AD = a\. IACB : IACD = 120" : 2 = 60". a'-2 2a'13

Aa = a :sin 60"

2a,li Ridnoeida::

JaB

cM.

ryra uixc nrrorrlrrraMu (AKC) i (ABK)' BM LAI{, rotii MC LAI{,6o IABK i *ACK - pieHo6egpeai a ocHonrroilInHaMlr (AKC) i (ABC). r'ow AK. IBMC - ryr uil+c

flosHaqumo cropoty piaHocropoHnr,olo rpl'rnyrsuna a'

! sc -- o., 22 ^ti . AD - AC.sin60" - "v" 2 DK

AD=

3 IADK: AKz

!g1 =o& .*. 24 o

BM'+MC'-BC' soso"= 2 EM .M25, AtBl

=.tIC' , AM'

6a2

-16

Eo"'

Vto

ou/io 4

roa2+ loaz 16 16 -a'

;r@!@ 44

a2 5a" 445

ct =

arccos;. c

npoexqia nigpiarca/-B Ha ll BBr.

AtBt

AB, CC, L a, C, - cepeguua - cepe+hsa (ea reopeuoro (Da.neca). Ba yuoeoro AA r: BBr= 3 : 5, AAr= 3x, BBt - 5x. rpaneqii frBAt' Ocni;*,nu CC, - cepe4ua,'rinia 1 Bq 4 !1" ; 4x=24;x=6. ro cc. = n2 , 24 =3t AAt=3 6= 18(cna); BB, = 5 6= 30 (cm). Bi}noei}a:18 cn, 30 cn.

714

:::::::

rx - )tu rc

pN)

B) =

1,b (cM).

NK2=TNz+TK2i

Nr =,[NK' - rK"

Y tMNT: MT = MK

!Eb'

L's' = !fu.5 l's)(B's (cm).

7:K = 6

- 1,r)

1,5 = 4,5

=,1,i

=.Eo.

MN' = MT2 + NT'; MN= laJl'F10 = Jn,rb +tO = ^FO,%= 5,5 (cu). BiOnoei1t: 5,5 cm.

27, IABC

trpflMor(yrHrfi pinHo6egpenuin: lC = 9O', M e (ABC), MB I (ABC), ntB = tli 'ELc*. Ocxirrrz MB {ABC), ro MB L BC; MB L AB. pieno6egpeuufi. IMCB - 45'. aCBM - npamoxyrHufi BC LAC, rovy MC LAC Ba reopeMorc trpo rpr treptreH-

s*"=|dc Bilnoei\u:

(cu,). swc

- niniiinuii lcyr ABorpannoro A

tE tE-1

""=; nli .*'.

lZ c*

=r:;=

(crnr').

aBMCI

- oproroHa,'lbna n.notqrzny u: AA, L a, BBr L a, rofi AA, C

BTATIOBIAHIZX TPIAfi YTIIIIITlB.

Kyra, y'rBopenoro trrou{ruaMu (AMC) r (ABC). 3 roro, n1o MB L(ABC), BlrnrrlrBae, u4o a.4BC - opToroHarbHa npoenqia rpurcyruzxa AMC. S"*. = S"*. : cos 45o. .-

DK2i

3a reopemoro Hoczuycin

a NrK:

Arnyrspu. To4t /MCB

3 IAMC: AC2 = AM' + MC2; MC

FD,

AC = BC =

zACD = 60";

r"*1.

= ADz

A-F,

MK )l BC ax ni4pisox, rqo a'e4Hye cepeAuurz rrporr4r're)fiHl4x cropiH xra4para. pirno6ivna rpaneqia, MN MK = 6 cttr. MPNK li 4iarona,rr,. .flpoee4evo NT L MK - Bwcora rpaneqii.

pe6po ABorparrHoro

^ACD: co

cepepnay CD.

*" =;FD - 3,5 cu nn cepe4ui .rriHii b

flponegeno

cepe4zua-4-B.

vr =f;dr = 8,5 crvr, NK

24. (ABC) f 6BC), rABC = r KBC - pinHocropouni. (ronn x me4iaHir) aABa i tKBC' /ADK - nisifiAD L BC, KD I BC - Brtcora nuii nyr ABorpaHnoro xyta e pe6pou BC, IADK = 9O".

pN I BC, pN =l=eC=B cu, Mp. 2

.A,I{

cepeglrna FC,

cu. F e (ABC).

IlossaqnuoP - cepe4zxy MF i K Poar.n.sHemo qortrpuryrHzn MPNK.

a,A€8.

flpoee4elno AD LaiAB LF. Il;rorqrzua (/3D) npoxo4urr uepea npsui-4-D i-4-8.

r{Ba,qpar ei croponoro 6

AF-BF=CF=DF=7cM.

(cru).

rEOMETPI9 Sincxixa O. 9. ra ix. Ara,qelurirHufi piBeHb liiiii:;iiliiii

28. MO L (ABC), MO = 3 cu ni4cranr ni4 rouxr M go ntot4uuu (ABC). Ocrci.nr,rcra ro.rxa M pinnoni4garena eig cropiH IABC, ro O noJra, 4enrp

Bnuca-

(nucor). Horo n a-4-BC, - Toqxa neperr{Hy ue4ian Hexaft AD ue4iaua i grcora IABC, rogi MD L BC aa reopeMoro trpo rpr.r trepneHArrnyJr.rrprr. MD ri4cra:ur ni4 rovru M go cropouz BC. MD = 2J3 cv.

t MOD: MDz

r.-

= MO2

+ ODzi OD = JMD" _ MO,

on = lbJs)" -l'z = h2-e = Ja A;le OD

paAiyc BnrzcaHoro

1cuy.

HoJ'ra.

3a $opvyroro aa,rexHocri paaiyca BTrr.{caHoro KoJra i cropoHn npaBl,rJrr,Horo rpr{}cyrHuxa a 2rJ3, roil BC = 2. rE = g (cu). Bi)noei0u: 6 "6 c*r.

[itilitilli{

TEOMETPI9 Finsnina O.

ra ia. Axaseuivxun pieenu

l{:i

Z1S

MOAy,nb 7. ySArAIbHEHH9 I C]4CTEMATt4SALllg B]4BqEHO|'O

{acruxa 3 Ilpaaunonutt. mempae7p

29. SABC

Ilponegeuo

3 oSOB:

pinuocropon-

SBz = SO2

JsB' - oB'

niAcraur

o"=#'

OB2:

- Mo"

.-, -a" s1n- cx

cos

3 rBMO: BM =

cl a sin o cos0 sinc

,(omnoxrzlto qlrceJrr'Hun "o"b

"'"i"m .in'o

cr cos oc acos' ct . =___, dsln-cx dsinq'slnct i sHaNreHHu( Ha o: AM2 a" a" cos'd *T

d cos

"itf "

7.17.8.

o (axc, IIr).

Tovxrz .A,

D CL

morxna npo-

n.noulr,rni ct.

i b, axi

neperr,rnarorhcs n

roqqi

N O(

oAHO3HarrHO Br,rSHaqaroTb fiJrorqrrHy cI.

,{ai rouxz A i B npannoi AB .ne*carr y n,ror4rani a,6o A e a, B e b, roMy 3a uac.nigxolr a axcioru (52.2) AB c o. Ananori.rao ArB, c cx,.

*"'" = Effi'

7.22. S. ' cos

Saaacni4norvr a arccioru crepeouerpil uepec rpr4 rorri C, ari He Jrerfiarb na o4nift npnnnifi, rroxna npoBecTr,r eAfiHy [Jrorqr,rHy g. To.rru A i B tencarr y n.noulrzui c[, roMy Bcs trpffMa

xn A, B <p'

+. ,5^

v[(18li4x1r16X1s - 6) -- ^tf8

repoua: p =

1^6

+2'n = J9'24'r'+3

= rs; =

i:::::

7.16.

npaui, Mae 3 nJron{ilHoro c 4ri cni.nsui ro'ur'u M i N (rovru fieperr,ruy B AaHnMn npaur-ruu). 3a uac,ri4xou s axciom (S2,2), sxiqo 4ri rouxr,r npamoi na.nexarr n.norqrni, ro i scg npsMa c Halrelx[TF,

7.21. Ilprrvri a

3'2.

716

7.11. r.

perr,rnae 4aHi

4h = zals @tr). 24\E J3 ; t0=:lO'. cosq=::r:= =

7.4.8.

7.5.4.

o 0) = cos o( ,

3uafi4enno nrrorqy npoercqii ea Soprry.noro

7.15. E. B.

7.20. Ilpone4eMo qepeo 4aui npavi a i b n;rorquny o, Ile uorrtna-apo6rarz sa arciol{oo IIr. flpanna c, sra rre-

r:eopeiuorc npo oproroHaJrbHy npoefiqiro S"o =

3ni,4cra cosq =

7.9.r.

r{epea nparvri,4S i CD, r1o neperr4Harorbcs B roqqi C, aa arcciouoro II, uoncna npoBecTr4 eArrHy nJroulr{Hy.

31. ,{Nn. N, 6.78. 32. 3a

7.10.

a cos

7,14. L.

BeBD,DeBD. 7.19.CeAB,DeAB. {epee 4ri roqrcra C i D sa a}cciolroro I,

---:sin(90" -

Bo = JEM' - Mo"

7.8.r.

BecTr4 eAHHy rrpgMy,

- "t"' ", 4 =\/E1lfi'il ""."

Eo=

7.13. A.

Ha npoBecrr,r e.4r4Hy nJrorrlzuy

a6 --*L. sln fl, 'AMO:

AO BO

7.7.8.

B, C i D,'relr(arb y r1ift n.norquui,6o A e AC, C e AC,

flosua'ifiMo fioro gonrxi4ny a' 30. MO I tr - nepner,qlarcyJr,Ep. noxu.ni, OA i OB MA i MB - ix npoenqii. Hexair IMAO = d, roAi ZMBO = 90' - ctc.

Jill'

7.18. Hexa* giarona;ri AC i BD vorrapuryrsaxa ABCD fieperr4rrarorbc,s n rou4i O. r{epea ABi npaui AC i BD, qo fieper]rHarorbcs, Mo]rd-

Ao =

7.3.

BiAnosiAb: ""16 cM.

7.1 . Ocnoeni Sirypn reoMerpir'ra ikne posniu4eHHe y npocropa 7.6.8. 7.12. A.

7.2.t.

r4SC.

7.1.A.

L(ABC\. O * tovrca neperr'rny meAian (uucor) aABC' SO

pa4iyc onrrcauoro r(oJra.

npanz.nrnnft rerpaeAp a pe6pou a.

si4 rourcz S go rpani OB

rpraryrHa nipauiga, y axoi nci rpaui

ni rpzxyturaxz.

rEOMETPIS Sincxixa O. 9. ra ix. Ara.4emirHufi piBeHb ::#:i:i:::iii:ii

AB c a. Ana;rori.rno AC e a, M e AB, K e AC,rouy M e u i ff e

Xe MK, rouyXe

7.23. Ilpnrvri

aib,

o,,

romy.MK c

crc,

cr.

axi neperr,rHarorbca n'rovqi O, o4uo-

0. Ocxi.nrrra roqrcra ,4 i B ira.nexarr np,fiMrrM a i b, ro Borrr Jrell{ar}, y n.norquni o, ToMy npa*ra.A.B Jre*cr,rrr, SHaqHO Bt4SHaqarOTb nJrotqrrHy

y n,rorqzni

cr..

To'rira Y ,'rerrcrrrb na npamifi.AB, rorvry Y

cx,.

[r:{l{iiiii{l$ TEOMETPI9 Eincxixa O. 9. ra in. Ara4euivnrafi pieenr

fiii;

217

lly lll)llMtrx ll I I'x llJllrll(rrr!l It llltnMv I t, rlrrlr!rr1'rhrrY

npfiMy, napaJlenrHy b. A 3l{atlurb, ltl)tlMtl ('r6itrttlrl,cf 3 rrpgMoro cr, a orlrce, re?tcffTb y uouluui u. Taxult uuuolt, 6y4s-arca np,gMa c, napâ&#x201A;ŹrJrerbHa b, axa neperuHae npfiMy a' Jlo)lcn'rl'

y n,rorqrni c. 7.25. [us.

aa.qaqy

J\! 3.38.

7.26.Hexafr ABCD - 4anrzft npocroponzfi uotrapzxyrcepe,4zHn fioro cropin. srax. Hexafi Ar Br Cr D,

AC, ToAiArB, cepe4tre,'riuia rpurcyruunaABC,-ArB, cepeAHrr il:nia IACD, Cpr AC. CrD, 3a oenaHoro napa:re.nrnocri npaltzx ArB, ll CrDr. Alraloriuuo 4ono4umo, u1o ArD, ll BrCr.

7.33. [iarona,ri xnagpara ABCD neperranarorrcs

I BD, AM I (ABC), AO L BD, rogi

voruptnyrrvr\ AlB

i iioro rporrz,rerxui croponra

tD1 Jrex(rzrr, e ogHifi

napanent'Yi.

n,rouluui

ABCD

trapa-

JreJrorpaM. pe6ep AS, BS, BC i AC - cepeAllHu Bepilu Hu flpocropoBoro qorrrptrnignoni4Ho. I{i rourcr naKyrHuKa. V sanaqi 7.26 6ytto Ac:BeAeHo' rqo MNPK

giarouali, roMy BoHu rrepareJrorpaM, MP i NK - fioro perilHarorbcff i ro.rxoro ueperuHy 4irarr,c,s nasni,a. AHaroriqHo AoBoAr4Tr,ca, u1o ri4pisxz, rqo e'e4uyrorb cepe,4[Hu pe6ep,4-B i SC taAS i BC, neperruarorr'cff B rii rrce rouqi. 7.28. !ns, MoAyrrb 3, Tecr 4ra camonepenipnrz

Hexaiic( d,bcB.

Ne 31.

y piaHux nJrorqrrHax d i F.

Ilpone4emo upflMy c ll b, srca treperr4Hae trpsMy a e rovqi C. ToAi np,sui a i c o,quooHaqHo BlrsHaqarorb nJ'roquHy. Biasmeuo ua npauifi b 4oni.nury rouHy B i npone4euo-ueper Hei npauy a, ll a.

3a osnaxorc napane,rr,uocri rrJrorqr.rn o ll P. 9epes rouny C wroxua npoBecr[ eArzHy nJrorrluHy, irapa-

reluy

Ilpone4euo inury npsmy d, rqo6 nona 6y.na napale,'rrna npsuifi b'i neperr.rnana npflMy o. Ilparui c i d ,rerxars n o4niii n,rouqzHi. Ilpama a Mae B HrrMn ABi crrirbHi roqrtrr, a orr(e, Jlexcrrrb r o4nifi n.norquHi. 7,3O, BepulznuAi B napanenotpauaABCD, ua -4-B ;rexcuri y nrorqvni cr.

a orxce,

BepurrHa C ne .nexurr y uiii. CD I AB Ba oaHaqeHHEM rraparerorpaMa. "foli. CD ll ,43 sa oauariorc trapaJreJrr'Eocri npcmo'i

i npa-

n.no-

lrlrrHt4.

7.31. flplrnycruMo, rqo npaui a i AB ne napa.nemni. To4i moxc.nrani ABa Buna,qftn. uuuo6imsi. Togi sa osnaxoro uuuo6i*cnnx trpflMux ($3.1) lpa1) Ilpsui aiAB

Ma a neperrruae nJron-lrrny, s sxifi rexurs

718

::i:::

rourli O:

OM L BD sa reopeMoro npo rpu nepneHAr,rKyrsp[. OcHirrrcrr npama BD neptreHAr4nyrspHa ABoM rrpflMxM,

rqo fleperrrHarorrcs, rrrrorqr{Hr (AMO\, ro BD L (AMO) sa oJHaKoro fleptreHAhKyJrspnocri npamoi i naorursn.

7-34.Hexaft piaui n.rrorqzuz o i p nepneHgzlcy.napni go npauoi a. Ocximxu

7.27. SABC rerpae.4p. "lo,txn M, N, P i K

7.29. Muuo6ixni npaui a i b,rexcarr

Ornre,

All(:l) ulll,l./tr'.lot,l,rM, () ,11y,11q11 rr.l,rrr,nlrv iiorrr 2lirrrr Or:rrirn,nn MA - M(:, ,r,o r AMC pillo6c4perrnii. lltr u.nacrueicrro trapalerorpaMa AO = OC, rcgi MO nc4iana, a BHaqrrrr,, i encora IAMC. MO LAC. Anaaoriuno, MO * n:,z,cora pieuo6e4penoro rpr{rcyrnr,rfta BMD (BM = DM, BO = OD\. MO L BD. Ocxi,rsxr.r npxwa MO nepnenAzxyJ'rflpna 4nou npairnum BD i AC, rqo nepeurnarorbcs, rrJrorr{zHn (ABC), ro Ba osnaxoro rrepneHAuxyJ'rflpnocri npnuoi i n.norqnuu MO L (ABC).

7.32.

/1. Artr',tr.ltr,:r'trr,ttcv ('M(,;l(llrr lllx,ll(( tll Il.Jll.l\ll (r.llltV

npaua.A3, rqo cynepevzrl yuoni.

rEOMETPI9 EinqxiHa O. 9. ra ix. Axa4enisHrt piBeHb iii:i;rLiii;it

uepea

Toqr{y npoxoArrrb eAlHa trJrotquHa, neptreuArrnyJrspna 4anifi npamifi, trJrorqrrH[ d i B neperuuaorr fipffMy a n pianrzx roqrax A i B. flprznycruMo, fiIo firoquxlr o i B uarorr, cni,rrHy routcy M. flpoaegerro rrJ'roquHy y qepea rrp.rrMy a i.,rouxy M . Ocxi,rrn[ trpsMa a nepleHA[KyrspHa nror{uAaM o i F, To y leperr,rnae flJaorqtrHrr d i F no trpsMilM MAi MB, nepneHnrrcyrapHnm npslriri a. TarcNm qrruoM, y nrorqzni y r{epea roqry M npone4eni 4ei piaui npami, nepneu4lrryaapHi o4uiii npauifi a, rqo HEMOXJ hBO.

Orrxe, nrorqznrt c[

B He

uarorr cni.ntnux roqon, ro6ro napa.ne.irrni.

cepe4uuaBC,AB=AC,DB=DC. - rerpae4p. M AM piniro6e4peHufi, - fioro ue4iana, ro1i AM Br4cora, AM I BC.

7.35. ABCD dABC

r CDB rasotx pinuo6e4pennii sa yMoBoro i DM fioro me4iana, a ehavr.rrr,, i eficora. DM L BC. Taxulr unHonr, nparva BC treptreuAr.rlcy,'rflpHa ABoM rrprMr4M AM i DM rrnouquwu(ADM). Ba ogHarroro nepneu4ur<ynaprocri npauoi i nrrorqr?rHn BM L (ADM). 7.36. {nn. Ni 4.15. 7.37. AN L(ABC). fl.norquna (/DN) npoxo4nrb ny AN. Toai (-4rN) L (ABC).

qepeo np,n-

FD L (ABC). If.norquna (FDC) npoxogurb qepea rpfiMy FD. Toai (FDC) t (ABC\. Hexaft BK ilapaierorpaMa. - IBucora BK L AB, EB BK Ba orHaqeHH,sM hepneHAnKy,r.spHocri rpffMol i trrorrluHu. To,qi BK L (ABE). BK L DC, FD L BK sa oBHarreHHsM nepneHArzrcyJrspuocri npar,roi i naorquun. Togi

BK r(FDC). Il;ronlnuu (ABE) i (DFC) nepneH4zxyanpHi npauift BK, roti (ABE) I @FC\. 7.38, [ue. N! 7.37.

!rs. nnacrunicrr a $4.2. i;;iittlii{ti TEOMETPIC EincHixa O. L ra ix. Axa4ervrivHufi 7.39.

piBeHb

i{d 719

$7.2. flepneHAlrKynep i noxrra Ao nnoulllHll, ei4craxi

* 7.57. MO L a, MA, i MB - noxlr.ni, AO i BO ix npoexqii. MO = 1,2 cu ei4craur rig rovru M go urorqrzuu cx,. AM = 37 clr, -BM = 13 cu. 3 "AOM:

Ta Kyru

y npocropi

7.40.

7.45. A.

1m'- Mo' -- li?'z -Lr'z =,fu? -rrrw -14 ,lr{As = = 5.7 = Bb (crvr). B tBoM: Bo = ^[BM" -fro' = JB'z -tr'z = b (cu). AO:BO=35:5=7:L. Ao =

7.46. D. A

7.41.

7.47.8.

7.48. B.

7.42. A.

7.49, Bilnoaila: 3.6

7.43. B.

7.50. B.

7.51. A.

7.44.

7.52. MO L (ABC),

rowry

Bidnoai}u: 7 : 7.

cm.

flrcrqo roura 3naxoAr{Tr,cg nooa rrJrorr{[Horo MHoror{yruuxa i piaxori44aaena ni4 ycix ftoro cropin,. ro ocuoBa rreprrerAtrr.yJrfrpa, npoBeAeHoro a 4iei rovxz Ha [JroqeHTp KoJra, BrrncaHoro B MHororcyTHr4n. rrIuEY,

MO LAC, MO L BD.

ZMAO=tMBO=6O". DB = ABJ, = 8.,0 crvr; OB = 4r[i

Ma=coso!,, I 60o=j3=

a.,E

porvr6, AC a BD = O. 7.58. ABCD AC = 16 crw, .BD = 12 cru. MO I(ABC). Ba yuonoro rourca M pinnoeiAAaiena niA cropin polr6a, rogi O

cm.

BIII4CAHOTO IIOJTA.

(cru).

qenrp

F, N, P,1( r:oqru Aoruxy rio,aa ,qo cropiu pou6a, ON L AB, OF L BC, OP L CD, OK LAD. To,qi MN L AB, MF L BC, MP L CD, MK L AD sa reopeMoro npo rprr flepreHArr-

Bi0noaidv: [,.

KyJrspH,

MN=MF=MP=MK=8c}.tr.

7.53, {un. J\! 6.69. 7,54, AC = 5 3 .AMC:

Poar.ngneuo ICOD. OC nepnen4rzny,rapni).

(err.rnercrxuft rpuryrnux). AC.-. twC = = 10 cv. cos ti0" cir,r

3 tMBC: MB = ,[MC,

JIn

a'?=

Joc\ oD'= "61 6' - u/ioo = ro "*. 11 (cm2). s^"^"=:oc-oD=:.8.6=24 22

Jioo - 16 =

t So.oo = : DC .OP:

(cm').

Bi}noeilr': !,. (|x)'z

4'z

= (6x)2 - (aJF)' ; 25x2 16 = 9 (cu).

- t6 = 25 AO=\6=B cu. 112

16 = 36r'?

- 27; ILx'

!1;x=l(r>0).

3 yAMO MO2 = \32 3 r BMo Moz = i72 -

3 a MAP MPz = MO2

Mo =.rmp' - op"

O" -2

24 lO

=4.g k*\.

OP2;

4's'

v6-

4-8X8

-4-8)

Js,%trg= 6,4

(cnn).

Bi|noaida: 6,4 c*r.

poiu6, AC a BD -- O, AC = 16 cu, BD = 12 cyr. (ABC).3a yuosoro rouxa M pinaorig4a.neua ni4 cropiH pou6a, ro4i O * ilenrp

BnucaHoro Kora.

7.56, MO L P. MA i MB - noxra.ni, OA i OB MA = 73 cv', MB = 37 clt. Ocxi.nrnr.r MA < MB, ryo OA < OB,

OA=xcviOB=7xcv'

OP = =

2So,oo

7.59. ABCD

Bi1noai0a: E.

ro.rxlr ,4orlrxy rcoJra Ao cropin pom6a, , N , P, K ON LAB, OF I BC, OP LCD, OK LAD. Toai MN LAB, MF L BC, MP I CD, MK LAD 3a TeopeMoro npo Tpr{ treptreHAurcyrspE. F

- ix npoexqii'

MN=MF=MP=MK;MO=2cll.

= 169 - x2. (7 x)2 = 1369 = 49x2.

PoarrsHerro aCOD. OC

oc=7.ac=8 cu: on=l-nn=6

pona6a

cu.

cD = ^{oc' * oD" = Js' + 6' = .r[oo = to

Bi4noei1t: 12 cm.

ii::: |EOMETPIS Sinsnina O. 9. ra ix. Ara.4ewriqHui piBeHb

LOD (piaronasti

nepnen4unyrnpni).

7369 - 49x2 = 169 - x2i 48t2 = l2OOi x2=25ir=5(r>0). Mo = tl6s -2b = ,1144 = 12 (cm).

72O

cu.

cD =

BC'

11^D S...-, = 1 MB. BC = i +.zrln= +rDi

(5 .

OD (4iarouaai pon6a

oc=L,+c=8 cu: oD=LBD=6

= 1EI = zvET (cu).

7.55.

iiliiiiiiiii

c*.

i:iaiiii:;i{f::i FEOMETPI9 Fincxixa O. 9. ra iu. AxaAeMi,{xr,ril pieenu

tt!;

721

"^".^

-oo = 1.8.6 1o" 22

V tBOC IBOC,= I8O" - 2IOBC = 180.

24 (cu,).

2s-"-:o 1 DC .op; op = ='-4 2DCIO 3 tMOP: MP'= MO2 i OP2; vp = ,[rwo' , ap' = Jr\ 4"8' = $ *

s^"^" =

4,8

(cM).

zs,o+ = Jr?

fi4 = 5,2 (crrn).

Ioo = 2x2 +

Bi1noei)u: 5,2 cv.

7.60. ABCD

IAB, MK IBC, MP LCD, ML LAD.- ri4crani ui4 rouxu M Ao cropin rpaneuii. OFL AB, OK I BC, OP L CD, OC L AD sa reopeMoro npo rpt4 nepreuAarcyrrspu. OF = OK = OP = OL nrc npoexuii pingnx noxranx. qenrp HoJ'Ia, BlucaHoro n rpare4iro, OF = OK = OP = OL Tarrau vranom, O fioro pa4iyclr. flpone4enro BH LAD - Bucory rpaneqii. BH ll KL, BH = Ocninsxu 3a yMoBoro r rpaneqirc Mo,fiHa Btrrrcaru KoJo, To

.ABH: BH =

oL xav

KL =! =:222

=!r*"o

AB.sin 60.

cv. AB

="f=

6Jd

KL.

= 24: 2 = 12 (cu).

n = 1136 = 6 1cm;.

3 IAOB AB2 = AO2 -t

Ao --

oB' = 'IAB" -

Bionoeiou:

100

OB2;

B cepeAr,rHrruM nepnen,qraKyJrEpoM, npoBeAerruu go

7.63. B ro.mz-A Ao nJ'roilIr,rur,r o npoBeAeHo AO i noxzli AB i AC, OB i OC

Eoo

1o!6

=10

(cnr).

nepleH,4anyJr.flp

npoexqir qux

I2O".

6ivnoi cro-

"[r-1d

BiOnoai1'a: 20 cm.

7.6.4.V yABC AB = 15 cru, AC = lB cu, CB = 14 cu. flponegeruo

cw.

.-'

BC, roni MK

npo Tprl neprreHAr,rriyJrspE. 3naft4erro S"*" B& OopMyroro 15 + 14 + 13

7,61. Tpaneqis ABCD Btrzcana B (oJro. Ocni;ruclr 4yru,

uix

naparelsnuMlr xopAaMrr, pinni, ro pinsi i xopAra, axi ix crnryrorr,. Ocriarxr.r BC ll AD, ro AB = CD. Tpaneais ABCD

BC aa reopeMoro

lepona:

pieHo6ivna. qeHTp onacaHoro xora. 3a yMoBoro ocrroBa Hexafi O BC crarye 4yry n 60", ro6ro 4enrpa.nsnwfi xyr BOC A 4opisuroc 60". OB = OC stu paaiycn orrucaHoro KoJa. Pinno6egperrzfi rprrnyrrir.rx BOC a rcyrou.60" rpu nepurzni e pinnocroponuiu:

BO=OC=BC=16cu.

s*.

'-=21'trxzr L<xzr ts - ) = ,i21.6

= v'5ttzt

.B =

rt.l.s.2.,.2.4:7.g.4= 84 (cru). Sinuoro6oxy S*" =it" AK, saisxu AK =25*"" BC 3 aAMR MK = JAM" + AK, = ,!tg * t* = zO (cn). =

2'84 14

(cru).

BiOnoei1a: 2O cn.

y yABC AB - 17 cryr, BC - 8 cu, AC = lb cw. Hannporu rrenuoi cropoHr4 BC lerrrzrr uennlrft

7.65.

(crvr).

Bidnoaida: 2O c*r.

pinni noxrari, BO i OC ix npoexqii Ha nrorqr{Hy cx. 7,62. AO L a, AB i -4C Ocxittxu AB = AC, ro- BO = OC. 3a yuororc BC = lO cv, ZABC = 6O", lO BC = 30". pieV pisHo6e,qpesouy a, -BC rcyrrr npr4 ocnoni 4opienrororr 60', rouy aABC HocropoHnift. AB = AC = BC = 10 ctu.

"*.

AM I (ABC), AM - 76

Ilponeseuo MO L (ABC), MO = 12 cv. eig rovxn M go eeprrunr.r rpaneqii. 3 aMOC: MC - r.i4cranr MC2 = MO2 + OCr; MC = "lTrz -*" = tT44+r56 = J400 = 20

100

to-6

poHr4.

722

(r > o).

= JM4 = 12 (cu). 3a reopeuoro xocrznyciri s a BOC: BC2 = BO2 + OC2 - 2BO .OC. cos 120.; 192 = 52 t x2.t 2. 5. r. cos 60"; 861 : 2b r f * bx: rc2 + 5x -336 = 0; r, = 16, xz= -21 - rre aa,4oBoJ,rbHfie yMoBy zagati. OC= L6 clr, 3 = Jn4 + 2s6= J400 = 20 (cu). ^Aoc Ac, = Ao2 oC2; AC =

I{exrp nora, orrJ4canoro HaBKoJro pinno6ivuoi rpaneqii, ire}fiI{Tb na nepetzui oci

gaxnoqesi

Ao = ,IEB, - oB, = u{FJ' = v!69

(crvr).

Bilnoei}a: 6 cm.

rpareqii

= tao; x =

rroxnJrux rignosigno. .48 = 13 cm, OB = 5 cw, BC = 19 cu, ZBOC 3 aAOB AB2 = AOz r OB2:

eJe t"*1.

=,!ont' u oL' = f,l' t (lG)" ='t0.

crauerpii

cvr.

Bx,

4o nnorquuu o,. Toqna M pisuosi4Aarena ri4 ycix

cropiH rpaneqii. Ilpoeegeuo MF

BC + AD = AB + CD

.!;

z*'

Bo=oc=#

pinno6ivua rpaneqia (AD ll BC, AB = CD).

P*"r,= 48 -ev, 1A= lD = 6O". MO I o. ei4cranr ri4 ro.rxu M

2 . 80" =

3a reopeuoro xocr,ruycis: BCz = BO2 + OCz - 2BO . OC. cos 120.. flognaqlruo BO = OC = x cw. lQz = yz t, ,z 2 . x . x. cos(180" 60.); IOO - 2x2 - 2x2 cos 60";

iitl rEoMETPlg Einqxixa o. 9. ra ix. Ara4evrivHr,ri piBeHb i:;l!ii;i;i,!iiii

xyr A. flpone4emo MA L (A_BC). MA = 2O cw. 172 = 289, 152 + 82 = 225 + G4 = 2a9.

Maewo: AB2

= AC2

+ BC2.I-{e oaaavae,

npauonytnzri. AC L BC, rogi MC MOrO trpO

i::ii:t{i*ii

u1o a ABC

M 2O cyt

t BC aa reope_

Tpu uepfieHAnr(yr.flpu.

FEOMETP|C Einsrixa O. 9. ra ix. Ara4eruivxui piaexr

i:ji:

Z2g

.AMC MC2=AM2 +AC2; MC=JrO" -15" =J40o+2%=16t5= 25

Ocrci.n*rz as = 2rJ3, 4e r = SO, naeuo: -4C = 2. SO. JZ = Z.OJE = fZJS BiTnoei\a: 1216 crvr,

(crvr).

Bidnoei}a: 25 cw.

7.66. dABC

7.7O. !un. N! 32 s Tecrie Arrs caMonepeBipxra, Mo4y.rr 6.

np,nMoryarvft (ZC = 9O"), AC = 18 cru, BC = 32 cru'

rinorenyau, MO L (ABC), MO = L2 cw' - cepeguua IlposeAelro OK LAC, OL L Bc,roai MK LAC i ML L BC Ba reopeMolo fipo rprr rrepflexAr{r(yJr.fipu' MK i ML O

ox=Lnc=?2=rc 2-"2'-"22

=\o"=4=s

cos60o Snai4euo nJrorqy npoexrlii aa $opuy;roro lepoHa.

.n'

^_!5*4+1Q_o,. '2 s,".= V?r.fzl -rsx21-1aX21- 1b)

tc*t.

KO'; MK = Jle

s." = 84

Jrse = 17

3 aABD BD2 =AD2 * AB'zi BD = $z'm -- JT44+289 = n[43 (cNr). Ocxirrnra AD L (ABC\, AC L BC, ro DC L BC

Toai BE

Ha Ao ocHoBrt, e rrne4ianoro,

='ltz' * ts' = Jl44 + 2%= 166, = Js - 4t =3141 i 3Gi cm. Bidnoaida: Ge3 "*

qerrrp Rora' qae, trIo ocHoBa rrepneunrryJrfipa MO BrrucaHoro B Tpzrcyrurrrc' Toqxa O nencr,rrr na neperuui ue4ian i nrzcor \ABC. OS = OP = ON - pa,4iycr KoJIa. ni4crani nig tourclr M go cropin MS = MP = MN

L AB, MN L BC' MS

Jnb

+ 64

.i{og = 18 (cu).

cv.

.rreHaAoocgoBn,cMeAiaxorc,

(cu)'

ro AE = I oC = * = 8 (cu). 22

E 16 cu

aABE BE =.tEz" - an" =,lt l" - e, =.@g 44 = Jzx = u (cu). Y aADE IDAE = 45", DE = AE = 8 (crvr). 3 a BED COSO(

Ba reopeMoro Hoczuycin: BDz = BE2

BE2 + DE2 *

BD2

15' + Bz -

2.15

2BE.DE cr

3 rinqin ni,4piexa

+ DE2

228 + 64

2BE . DE - cos u;

- i,69

15.16

L2o 1 15.16 2'

= 60'.

nepneHAtrr{yrspni, c - npamaixneperuHy. Ae a, B BC L c, AD L c. Toni BC I a i,4-D 1 B.

-A-B npoBeAeMo

AB = 24 cM, BC = 12 c;ltt, AD

i:{iili:iii*i

].B2

Bi]noeiOa:6O'. 7.74,lltrots1uwa

= o t"*l'

piBeHb

l3 cu

L ra ix. AxaAeuiqHrlfi

-2.15.8

zBE . CE .. cos 60o;

ocuonyAC,AC = 16 cm,-48 = 17 cm, BD =13 cw, ZADC =90". flpoxegenro qeper rp,flMy BD n;rol4zuy (BD-O), nepnen4ranyrr,rrpry go npruoi AC. Toai BE L AC i DE L AC. IBED = cx, nyr ui}lt u,rorqunannz rprzxyrnnrcin. A Ocxiruxu r -piauo6egpenoMy rprrr{yrHuxy Bzcora, npoBe-

pinuocropoxnifi. Toqxa M pinnoei,44a:reua ni4 (Arae, pfic. A9 )l! 7.68) aABC ftoro cropin. flpone4eIvIo MO L (ABC\ i MS L AC, MP L AB' MN L BC, AoB'r{rHr'I qnx niApisris * qe ri4crani aig tovrcra M 4o cropin a-A-BC' Ocxi'nu(r4 3a yMoBoro M P = M N = M S, ro OP = NO = OS src npoexqii pirnr'rx roxr{rrr{x' OP L AB, ON L BC' qenrp Ho'ta' Bilucarroro B a'A-BC' OS LAC Ba reopeMoro rrpo rpr fiepfleuAtrrtyJrfipra' O

rEoMETPlg sincrixa o.

152+82

7.73, Pirno6e4peni rpnxyruaxw ABC i /J)C nnarorr cni.nrny

7.69,

724 lilj

=!2 AD = 8 crvr.

- AE" =Jl?, - 8' =ft?-8)(1?.8) =J9.2b =B.b =1b (cu).

Bi0noei6a: 13

Bi\noei\a: 4Jd cu.

6f 3 aoMS Mo = 6{3 cru,' so = +2= tg60' Ja

mirx

aACE ZA= 45',6o rpraxyrnrax pinno6egpenrzfi npanoryrHuft.

BC=

f t; (rs=r= AB 12 =9!9=2J5 (cu). 3 2J3 2.13 r" a MS2 = MO2 + OS2; MS = Jo' * (26)' = .,60 - 1, = J4s = 4rE

AE = ED

3 r BCE Ba reopeMoro xocranycin: BC2 = BE2 + CEz

TeopeMoro npo Tpr{ flepnerrAr4rtyJrfipi'r'

IMPO=ZMNO=IMSO=60".

JB'

EC= M =9=8 t".r. ts45" I

(cm).

7.68, yABC - pionocropounift', AB = BC -- AC = 12 cur' To.nca M pinnoeig4a.nena ni4 cropin aA-BC. I{e oaua-

TPITICYTHIIICA'

^ARE:

BE-

I AD, CE L AD, ZBEC = 60' ryr -

urorqrzuaMrz (ABD) i (ACD). Ocximxu n pinuo6egpenouy rprftlnrrnr{y Bucora, npoBeAe-

3a TeopeMoro ripo Tpr,r nepilerrAr4rcyJr,fiprl. 3 yADC DC2 = AD2 * AC2i

6O MP

.72. Pinso6etpeni rpranyrnrasu ABD i ACD uarorr cnilrny ocrony -4-D, AD = L6 cm, AB + 17 cm, ZACD = 9O'. flpone4euo qepes rrpsMy BC n;rorquny (BCE\ L AD.

(crvr).

= 1G8 (cu'z).

0,5

AD L (ABC), AD = 12 cw.

B yABC AB2 =AC2 + BC2; Au = .,85? 8' = Jns + a+ =

=,1rr.8., $ = js.?. +2:r.T2

=3.7.4=84(cm).

[44+256 = J400 = 20 (cru). MLz = Moz + Loz: ML = ^l{r\r" = Jt44-el = tTrs = 15 (cu). 7.67.V yABC lC = 90', AC = 15 crr, BC = 8 cru. D MK2 = MO2

S'o

7.71. S.=

ni4crani rig ro.tcn M 4o rcarerir rprrriyrrrrrfta. Ocnimnu ABi flpsMi, trepneHAllltyJrspni rperifi, napa- A .ne.rrni uix co6oro, ro OK ll BC, OL ll AC.To4i aa reocepeArri trittli rABC. perroro (Dareca OK i OL (cm);

(crvr).

=l2Ji

clltr.

i:ii:;ii:*i:;ii fEOMETPI9 Einsnixa O. 9. ra ir. Axaseuivnui pieenr

iil;;

Z2S

3a osna.resnnrvr trepfieHAultyJlspnocri npauoi i nnorqr.rHrt BC LAC i ZABD ftyrr, yrnopeni nigpiarort.4-B a n.norlunaMra o i P'

iAD L BD. IBAI

B IABC AC =,[Au" - Bc" = J;4' -Lr' = Ff-tzl<z+.rr) =,M.86 =6.2Ji = 12J3 (cM). X+=

,:rru=5.

,'"o"=ro".

,4o sag:aqi

rTeperlrry.At o,Bc

.74) fhorquHu o

nepneugunyrapui,

npflMa rx

cwr, AB = 4 cm, DC = 4 cm. npoexqis oi4pieHa-4-B na nnorqraxy crc, BD Ha n'norqr'rny B' yrBopeui ni,qpiaxou -43 a n'noqznauu cr i B' llyryn BAC i ABD - nyru, B IADC AC = ^[AD\ Dc' - J4" + 4' = ali kw).

3 "ABC LSZBAC=

4'[?

4J2

= 1:

IBAC =

tcBD BD = ^l6c\ BC' = ,l+' +(+"li)' = JI6* n = J48 = a.J5 AD4- l-, IABD - 30 . B \ABD tezABD= B- - 4Jg= JB

n= 8

= CO

cut,

(cm,).

='ff , o" a, b, c -cropoHrr rprrrcyrurrca, *o'u.3"i.q"" .:0"-='*^Uf o=2,b (cm). ' R - 4S-"'

B a OSB SB2 = OS2 + OB2; SB = Bidnoaidu: 6,5 cn. cx

2,b2 =

.,!62 +

!86

aR =

OB-

paAiyc Jrr""uro"o

6,rb = J4r,% = 6, 5 (cu).

npoBeAeuo neptreHAr4nyJrff p AO

AO = 12 cv, AB = 13 crr. flpone,4emo OC I AB, rogi AC

- npoerqia nepreEArxyJrflpa

noxnny AB.

AO na noxutry AB.

B aAoB oB =,tqn, - nO = lTB, -n, - b (cu). Hexafi BC = x cv, roni AC = (18 - r) cu.

3 a BCO CO2 = OB2 _ BCz; COz = 52 , x2 __ 25 _ x2. 3 IACO CO2 = AO2 - ACz; QQz = 122 - (13 - x)2

4b".

S*"

19. 3 roqxr A Ha rrJrorqrrny

4Ji

=!SC

qenrp r(oJra, orucauoro nasnoro a43C.

.,[6+4 =Jn=2r,6(cu).

A;re

iB-

4 cyt'. - nucora,AD= a ABC, rogi AO BO

* BD? = TV;,, _

s^,"" 22 1 Bc . AD = l.n

3 xinqin ni4piena /-B rpoBeAeMo BC L c, AD L c, rol-i BC L a, AD L $' BC =

Bttcorai nregiana, Zn

3 aABD AB =,,8D,

Bilnoei1a:30'; 45'. ,75, (\uv. puc.

npoeHqii pinuux noxz.nzx. To,qi O

SO=6cm.

3 .ABD an = Jl,n" - l,n" = -ltzJz)= ,,i76 - 288 - J2s8 = t2Ji @v). ADtp. .ABD = - r. IABD = 45". BD=t'9 12J2 7

pinnp6eApennil,AB=AC, BC= 4 cm,AD

SO L (ABC). 3a ynonoro ro.rxa S pirnonigga,reaa ni4 nepruzn

tgZBAC,

lB. IABC

= t44 - 169 + 26x x2 = -x" + 26x flpapieaacmo o4epxaHi Bftpaar4:

(cu).

Bi|noei|u:45'; 30".

- x2

.26

-x2

26x

25.

25: 26x = 50,

50.

50 1,"*t. = AB-.BC= 13- 26- tt 13 I c*. Bi}noaiila: 11 AC

Tecr Ans caMoKoHTPonK) r{acruna

1. E,1..(. V IIepeTr4Hy 3

suna4rcy

I rpn rovrcu

Jre]Karb Ha

ognifi npgviii, oA"u

Apyron fip.sMom.

1. E.

n'* -

rorrxa

14' r.

2. E.

5. r.

8. B.

3. B.

6. E.

9' A.

12' f.

15. B.

4. R.

7. t.

10. E.

13' E.

16. A.

Hacrl,/,Ha2

lC = 90", AB = 10 cu. 17. rABC - rpnMoltyrHrfi, Ocrcixrrr roqxa S pinnoei44a.nena nig neprrzn 4,43C, qeHTp r{oJra' olr4To ocHoBa rreprrenArrnyJr,fipa SO canoro HaBKoJro nroro. V flpsMoxyrHoMy rplrx)nH[Ky nigcrats nig ro'rxu O cepe4uua rinorenyau. SO

O go n,'rou.1zHz rpzr{YTHrlxa. 7? Y ^Aos ao =l 'qr- = o cv; AS =

c:vr.

22 so =.lTs' - ao' = Jlo' 4'-./oa - a c..

Bi0nosi}u: 8 cw.

726

ii::i rEOMETPIS EincHiHa O. 9. ra ix. Axa4ervri'rHxf piBeHb i!iiii;:i:':+ii

(ABC). 20. ABCD - npsMoryrHlrn DK L DA IAB Ba yMoBoro, rogi KA I AB aa reopeMoro rpo Tpr4 fiepneHAr4nytspu. KA ri4cranr nig rou:cn K 4o croponra A3. KA = 2,4 cm. Ana.noriuno, /{C - ni4cranr ni4 rounu 1( 4o croponu BC, KC = 2,8 c;ltt. I{D - 3,6 cu. 3 IAKB AB2 = KB2' KA2 = 3,62 - 2,42 = = (3,6 - 2,4)(3,6 + 2,4) = 1,2 . 6 = 7,2. DCz = AB2.

B rDKC DK =.tEC Bidnoai4a: O,8 cv.

- DC'

$,8" -7,,

Ji,84JJ

= J0"64 =

0,8 (cu).

21. Y rABC ZC = 90", 14 = 3O", rcai lB = 6O'. AC

6un;smfi Harer,

aexzrr npotz 6i-nruroro

rocrporo xyra. AC = 6 c*r. KB

K 2JB a*

(ABC),

KB = 2J6 cut. Ocxi.nrnlr BC L AC, ro KC L AC aa reopeuoro trpo rpu neptreuArrxyJrspu, KC eigcranr ni4 ro.rxra K 4o rarera AC.

l{lilit:;ilit

TEOMETPIC EinqHiaa

0. 9. ra ix. Ara,qenirxrail pieenu

iiii:

227

3 aABC BC = ACtgSO" =6

BacBK KC=^[KB\BC' Bi0noai4a: 6 cw.

.6 2JE =

(cu).

-JGtr;G.fa'r =$4.n=./so=o

22. [:,as. ].1! 6.79.

23. Y yABC AB = 65

cm, BC = 75 cw, AC = 2O cw.

6i.nrrrr,rft xyr' 6i.nrrua cropona, rovy /A MA I (ABC), MA = 60 cv. flpoae4ervro -A1{ L BC, M K L BC aa reopeMolo rpo

..._

Tprr nepneHAr4Ryrtrpr4. nepnen4rrxyrrspa AK i,MK - siAcraHi siA xiaqin MA 4o 6inl-rtoi croponra BC. 3uafi4erro,41{ a tpoprty.nz n.nor4i a/BC:

Ilnoqy rpt'txyrurrr(a snaftgerro 5.

,a,x =2

B-

= 16

lepona:

65+2O+75 p=-=au.

B .AMK MK = "[MA' * AIC = Js600. = ft856 = Jl6'% - +J241 (cn). "56 Bi|noai\u:16 cu i 4J241 cru. S =,4-B . KP, ge AB

fioro ctopoua

cropona, a KP

- 12.cw. ancora pou6a.

KP=+ =lo (cm). AB =ry 12

op-!xp=5 2

ni4ctanr ri4 rouxra M 4o croponu pott6a, MP = 13 cu. = [6, -% = IA = 12 (cu).

Bidnoaida: 72 cm. ].1b

,43=5 cu i

o npoBeAego rreprieHAr4r(yJrspAO = 4 cra i Aei noxra.ni: AC=4JB cla. OBiOC- npoerqiiquxuoxu.nr,rx' Z.BOC=6O'. n;rorqrEny

3nafiAe*ro ix Aorxrnr.r Ba reopeMoro Ili(paropa s r:pranyrnzrcin AOB i AOC.

Bo = JAB' - Ao'

=,lrs -t6

= J9 =

(cn{).

* Ao' = J8o -16 = uGZ = a

oc = "t@ 3 t BOC aa reopeMolo BC2 = BO2

+ OC'

lcru).

rcooanycin: 2BO . OC. cos 60o;

2.3.8. +t = Jno = z 1cu;. Bc =,ls \l + 64 -1

Bi1nosi]a:7 cvr.

728

AB6=

= ."6a =

3J5

= -\6

1cu;.

: IABD = to'-

flaorqranr,r oc i B nepnen4ury.napni, CD - nparraa fiepert4ny nnorqnn.,43 = 25 cm Aaruft siApiaon, A e B, B e c. AC L CD ..._ nigcranr nig rouxrz A ,4o fiJrorqr,rHra u, AC = 1"5 cvt. BD L CD - nigcraar eig rouxra B 4o naorqr.rnlr B, BD = 16 civr. 3a reoperroro 8 (uo,4yJrr 5) AC I CB, BD L AD. g Togi CB i AD npoexqil ni4pisra -43 oi4noni4no Ha nJroqunlr o i p. 3 *ABC: AB2 = ACz 't BC2; nc =,[nn' BC: Jrs" -rb" = JZ00 = 20 (cu).

aCi

B oADi:

ABz = AD2 + BD2; AD = "[AB" _ BD,, AD = Jrb' -16' = J625,286 = .,,66e = sJal (cu).

- npsrror yrma, ZA = lB = 9O". AB = 24 6icexrpuca Hyra B. CM L (A.BC), CM = 7JG cv.. Ilpoee4euo CK ll AB, toxi CK L AD. ICBD = ZBDA sr< euyrpiurni pianocropouni. pi'nno6egpeuraft s ocHosop BD. To4i IABD AD=AB=24cw. B 3 aCKD (CK = AB = 24 cv): CD'z = CK2 i KD2; (cu). 24 cv KD = JcD"

6.81.

26. 3 ro.rxnA ua

ts "q8.lBD

29. Tpareqis ABCD

,B aMop Mo =.[Mp' -op' =,lls' -b'

25, llz.a.

JD, - Jn4 = v[S = B.,E (cnn). = Li ZCAB = 45a.

^tED"

Bi|noeidu:20 cnn, 3i,[T cu.

cu.

rcyru, yreopeHi nigpierorvr:43 s nno-

r{acrrxa 3

To.ma M piruonig4arena ai4 cropiu pom6a, rolry ocBnr4caHoro KoJra HoBa nepneHAuxytapa MO - rIeHTp O. Padyc rco.na r 4opianroe no.noelrHi Br,Icoru:

MP LCD {ea TT[I)

BC =

28.

Ba OopMy,roro

12O cutz, a

B aBCD

Bi0noai0a:45'; 30".

("*).

24. ff.norqa pov6a ABCD

3 a ABD cos.lABD =

* b., "B r. s = 4. E. b. 2. B = 600 (crra).

'6oo

uquaamlr

AB' = JAC'z.'+ Bc'z =

s*"=@=J8o.1b.b.6o= =.ll,r.

crvr,CD=3cru. Ityru BAC i ABD

3 o ACB te" tcAB = E9 =3+ AC 3J2

l-qx .BC. Ar( =2s*' 2BC

.^- =

1cu;.

27. IltrouTunu cl i.B nepueu4uxy.napni, c rrp,fiMa rx nepernny. A e s, B e 3 rcin4ir niApiaxa,4B npoaegemo BC L- c, AD L c, rogi BC L a, AD L B. Bilpiaxir AC i BD - npoex4ii ri4piexa ni4nori4no Ha nrrorr{r4nn a i $, AC = 3rE cm, BD = }JS

t::# FEOMETPIS SincHixa O. 9. ra in. Ara4er"rivHrfi piBeHb ::.:iiti::.:i::::i

-cK" -'lzs" -r4' *

="v[g

B aAMC: AM =.tmc' o Bi1noeidu: 4O etr.

CD = 25 cvt. BD

-t

= v?Bb + 86b =

172

= 576 + 289 = 865.

vt600 = 40

(cM).

pirnoctoponnifi rpltryrHr{r{, 30, Hexafi ABC - 4anzi cropoua rxoro 4opienroe 8 cu..43 ca, C e a. CD L cI, nigcranr ei4 nepnrraHr.r C 4o nnorqunra cr,

2JB

CD = cnt. Ilpoae4e*ro CK LAB, ro4i DK LAB sa reopeMoro npo rpr,r reptreHArrrcy.nspr.r. ToAi ICKD .niuiftnr.rft xyr ABorpaHHoro rcyra ui:*c IABC qlruo fioro g.na apy.rnocri g.

7 = 17 (cm). = AD KD = 24 BC = AI{ = 17 cvr (ABCK nparrrrorcyrnran). 3 rABC = 9O"): ACz = AB2 -t BC2 = 24,

(lB

cryr,

rrJrorrlrtuoro

c 2JF cu

c. floana-

:iiii:;ilt:l! rEoMErPIe Eincxixa O. e. ra ix. Axageuivnun pieenu

if{

729

3 ;ACK (1A = 6O"): CK

2"b

srKCD: 31

- AC sin60' 1. a 2'

"inw=ff= 4.rE

.t; T

a.,E (cu).

-- 30'.

. ;ABC - pinHo6eApeHnit; AB - AC. Po o",, = 128 cm. megiaaa (i nucora), AD 32 cm,

M e- (ABC), MA

MB = MC = 65 cw.

nig rovnu M Ilpoeegemo MO L (ABC) - Bi.4craHb Ao nroulr,(Hrr (ABC). To.qi OA = OB - OC nn npoexqii A qeHTp Kora, ofir{caaoro HaBpinurax noxurux. O xoro a,43C, AO = R. IloaHaqr.rNro CD = BD - x crl.

124 2r AB '-"- --' = 64 r (cM). 2 AC2 = AD2 + CD2; (64 - x)2 - 322 + x'!; 4o96 3 "ACD:

'loti

AC =

728

x2 = lD24

tc'!:.

l28x=3O72:x=24.

CD = BD

S.,"

-2

24 cm, AC

3 rfropvvnH "4Rs = AU

AB . BC

= 4 S.-.u,

3 yAOM: AM'1

aM =,[AM"

- Jsooo

BC . AD -- 24

ob:

rHaiiAevo 40 .40 .48

4 .768 = AO2

24 = 40 (cm).

.32 = 768 (cu").

. R:

. Zb (cMr.

+ OM2;

-Ao'=,[ss' z'; =J6b-r5X65+r5)=v?0

= 6o (cm).

Bi)noai)a:60 crt. 32, fhoquua, flKa trpoxo,ql4Tb qepe3 npflMy a i rtapa,rensHi npaui,4A, i BB,, nepernua€ nJronluHy tu no npauiri A,Br. trapaA1B1 ll AB, AAt I BBt 3a yMoBorc. Ton' ABBI, reJrorpaM.

AA) -- BBr - 14 cu, Ats = AtBr - 13 cpr, ArB = 15 cu. Srr,"r, - 2S*,r^. SHafigemo S*,u' 3a @opuyaoro lepona.

_=27n= l3+14+15 '2 s^^"" = 1Fr1n

-rslzr -raXzl-rsi - Jrlfl?€

-.tF.,vr:,.r4

=7.3.4-84(cu).

S-,r'n=

Bi)noai}a:168

7gO

;::::::

168 (clr'z)'

cm2.

FEOMETPIS EincHiHa O. 9. ra in. AxageMi.{Hufi piBeHb

i:i:i:i:::i:i:i;::::i