9 - Tacka, prava i ravan

Page 1

Ta..ixa, npaBa, paBaH Eyrnup ut Arercau4praje je4aH je o4 HajnoaHarujax MareMar '.iapa aHrtlror 4oba. Taj nelurc.l yulre.ru MaTeMar Ke ayrop je lyseHor cnuca EneMeHau, Kojh ce cacrojr o4 '1 3 KFbhra. Eyr<n14 je ocHroav reouerprje r(ao Hay(e: 4ao je

rarourrocTk x npaBxna y reo,uerpljr.r

xoju aaxe r 4aHac. Fberoen EneMeHnu, xao yq6eHax r: reomerprje, npoyuasajy ce cKopo 24 BeKa, npBo Ha rpLlt(oM je:nry, :arnm Ha apancr(oM, rarl.tHcKoM, Kao r Ha ToToBo cet,v ruo4epHlu je3rut4Ma. Eyr<nu4 ce y roM Eeny

6asro ocH o saua reo,raerprje y paeHr npocropy. l-beroaa reo,raerprja obyxaara o6,rare xojn ce Mory KoHcrpyacau nomohy aeuapa l.i r.lecrapa r npo6re,r,re H

r<ojr ce ,raory

peuarl

no,raohy

Ta ABa IHCTpyMeHTa.

Og rpnHaecr EyKnliAoBt4x t(Fbt.jra npB X uecr nocseheHo je reomerpaja y paBHk, loK cy nocrellbe rpl t($xre nocseheHe reomerpnja y npocropy. Y ocrarrr,,l r(tsl..iraMa Eyr<r14 paluarpa a prrMernLl Ke npobreue.

Eyxnug Apxrr',le4 u: Cr.rpaxy:e 4pyrn je Be,nrKt. 3Hanaq reor'lerprje. l$eroe pa4

o4srja ce y HenocpeflHoM l(oHTaKTy ca Eyx,r ugoaou uiKonoM. nope4 ocralor, raj r',lare,raarruap nocrao je cnaBaH y o6.nacrr reouerpuje no roue uro ce 6aeao npobrema,raa rojr Hr.lcy peureu nyreM KoHcrpyt(qrja rerutporu a uecrapoM.

rr

MareMaruqa pr, noacraKHyrH paraojer'a acr poHorurje

Spojna aHruv

I

MexaHrKe, baelau cy ce npobleuuua reouerpuje. Mefy rurua ce cl,j qe Apxur,,,le4

Ano.noH

u

je

r:

llepra,va.

AnonoHrje

Eyrnug, Apxa,raeg I AnoroHrje, c Apyr M MaFbe no3Har M MareMarlqaprMa, car(ynHnr cy, 3anucarH n objactlal cBa raAaulua no3Hara MareMarkql(a 3Harba. Ta gHaFba npey3enx cy ll3 apancKkx t43Bopa 3anaAHoeBponcKH HayqH q| y paHoM cpealbeM BeKy, Lr-rro je 6ao h3y3erHo cHaxaH noAcrlLlaj la pafarue MoaepHe HayKe.

48


t a

aUa

Koi4o nparhx oApeiyjy

aare raqk.?

,

Konr{o noooji npa8,r Koje carpxe ractry Ko,rKo no.rojr npaBkx rcje calpae Eqxy,4

troj. cy HopManHe3a npaly 4?

rcF cynapadHe

c

np

or

d?

r

(oja oa nyBr Ha d,ur a ayHiHyjen|ary paooja6y BMeiy racc A I npase,l

KoHcrpyru{ rajeaHrq(yHopMatryi

a 3arrM

EMepr r 3anrur y

wrrMflpMa

padojaBeddexyn srnpd*r.

@

ro;. oo;.*.

y;*"o; p"*,

oapebyjy cse racrc Ha {croM pamjaBy oa aare npaBe?

"O,,D7 UEJ"O "'O -'[n'E]

h "(l h^


Taqka H nPa Ba. O.aPebe\ocr npaBe. Taq(a

qr;* Tawa je oaoBHtr reoMsprjc(tr o6jehr. Tacry Hajqeuhe upraMo kao npalH kpyain, rcnybeH kpyxtrh hntr a8errHrje rcje .e ce(y. Taqrc o6eneaaeaMo 6en(( MnarfHNKft cno8 Ma:n,3, C..

*. npaoyrrHrjy

koja

je.a o6e.rpa|e HeorpaHNeka o6je6r.

{arhoaMo

nDaBa je ocHoBHr reoMerD* jc(tr

npa8ydauprexynp*63yjedoqprajynh jenaHseHAeo Toje npaBe GHaqaBaMo ManrM nrcaHr

i

'''-'--'''--.---

Kaaaraqka,a iptrnara npasoj p, 3anrcyje{o: A ep; €aa rdcKa 3 npaBojp, 3ai,cyjeMo: 3Ep.

hxero

He

nptrnala

aa .y (ontrHdpHe Ec€. jep C.y kofu Heapre nprna!ajy nDaBoj p.

jeaHoj npaBoj

50

, ,'"..

I


-! v oo4xHtr noooje

pdHr

o6je

m

da

(oj#a

-rlrrnllln

MoxeMo

yoqfin ipa@ n,Hrje.

-r_I;ai

Nauprajap.Byro6enea* je.ad.Hauprajaseraqke,,4trs, 6jenptrdMaiy npasojd tr ae nea, c I o, rcjejojHe nprnalajy

a)lbntrko npasrx Moxeu Aa Hauprau (Dd rac(y,41 6) (o^r(o ipaeu MoHeu !a Haqprau (porracKy 3? s) Konf{o Momeu ad HaLrprau npaBtrx {oje capxe rea8e rac(el

npasaje oapebeHa aoeMa parrrcFfMEcGMa npaBa koja caapM $e paviqheraqRe.

nmrojr GMo jeaHa

AB

@,, 6)

o!peteHo ca rph {erc^rNeapRe oapeieRo ca qertrpr parntrcne nooojerptr roje cy {o,xreapie?

I ;

HaLrprajHekon{HeapHeracre tr npaBe

(oie

GApxe

no.MarDalnKr nanrpa.3.MroH aa Djiro nooaje Ge rentr, !a co oa6oj(auo repeH, rao oaahoH, aepoapo trjou iHdo, MRorc mhi. Ta{o 6trcMoao6rnH pasaH. ue,a pasr He Moaeaa ce Haupr,, (ao uro He Moxe aa ce Haupra Hr qe^anOaea,antrMoeeceiaupra ieaaHAeo pa8dH.Iajaeo

je Berrktr (ao non ycroHrue,

Ha:r8a ce Moae, pasHf, a y beMy Moae0,4a

qprau

oaer npaBe

EE[] 51


PeaHje ocHoHh reoMflpfjcM o6jekr. PaBaH je HeorpaH,qe|a ca c,x crpaHa PaoaH oOeneHaBaMo Mantr croBrMa rpc(or afOabera: d, F,y,5,0,€,a,n.

lGaaraqka,4 nptrnaaa paBHtrd, sanr.yje denflMAa DsHtr d, rnmvle o:3 e0.

-----

'3/-

t

./

,/ --\

o:/ edj

(aaarac{a 3

---:------

n *\

3 \

\s1

D{e koje npfiaaaijearoj pa6B, uxeMoaa cy rcdn,aHapHe, a:aracrc (oF m nphiaaaijelHojpeHh GxeMo aa cy HercMn^a{apHe. HercMn^aHapHe raq (e npinaaajy npocropy. 3a

vo(oirHr Mo*edoyoqrn

farrprajpaoaHduo uro

I

c (oje

n

naaajy pa8dh

pa,He

oojere

Ha

KojrMa

npmo ajeMoMoae

cy oHe Ha nperxo!{rM qprexf Ma. HaLrpraj raq€ A, dr rac (e D, Er Frcje He nptrnaaaiy pa8Hf 0.

PsaH je o!peieHa rpfMa se@{seapHhracGMa

nmojh BcHojeaM

p

aH(ojaoapHtrrph

PaMH rcja je oapeteda rpfta HerorrHea pHiM raqkaMa .4, ar C NaioaieMo paBaF ABC.

52

e


-l CsrMo

ce oHora

uro.Moyqrnr

o

{!aapy.

KBaaap je aeo

ipocFpa orpaHrqeH npaBoyraoHhqrMa, re npaBoydoH|rc. le npa!oy.a4rKe ha3rbaro npaHaMa R!a,lpa. TeneHa npaBoyraoHKa 8a3rca{orseMMa $aapa,

vbycyjyhr r

a.rpaHrue hruaMa kBaapa.

Ksaaapha

8r

eHa,12

iBrua|6opaHa.

orrN je (Mnap ABCD jB,c,D,. lle.oBa FMeHa cy: r, A, C, D,4,,3j, Cl, D,. Ha

npftep,

Ha

crpaHe $aApa cy npasoyraoHrur: ABCD, BCC P I AP 1c p CrpaHrue npaBoyraoHrtra cy NBrue $anpa: 43, 3C, Co, 43,

I

Arc .v .rpaH{ue csfx npaloyradBa rcjo oDahrcabi RBanap Mebyco6Ho jeaxa(e ao6hjaMo aoMerpNjc(o

F

o koje ce Harhla KoqG.

Clpaie {ouke .y Rsaaparr

cA,3, c, D, t,,3,, q I Di. @ *.^,;" -*, *;" *o reMeta olHaqmr

racKa D nphaaa pa8$ kojoj nprnaaajy @(e,4, R Koje reMe (oq{e nptrnaaa paBH( o! peieHoj rac kaMa A, 3,

Aard

KojereMe{oqKe npina,ca pasH{ oapeiercj raqbMa.a, c Koje rEtruecanpxereMer?


7 .Eee npase.

l(oje ynrle.eKy

OapeleHocr paeHu

yiruy

Bace

qapanihal

Koje.y yirue napanenHe c lGer MrxaMmoM yr*loM?

"{\b

t -t

"',-"t

\;."*-l^

!RF

ftarc

ABe npaBe ory!a trMajyjelHy Taaa kaxeMo aa ce npaBe.ery. Na

np&ep, npaDep r

4E nese rcF

@

4 Ha

rjeaHHcryraqky.

otrLrH.e.eky

crv iprianajv

jelHoj

Da

#r

,q9e npaDe Moryaa trMajy.Be raqrc rajeaHNke

Taaa uReMo

aa.ere

npase nomanajy

fa npftep, npaDe a tr, Ha cirutr ce nodaiajy, 4E npare Moryaa HeMajv sieaNrckrr rc a Iaanp naaat kaHolpsi, Ha

npftep, npase. r

d Henajy

iajelHrclrx raqa€


!jeaHrqor sqac I nptrmnajy jenHolp Hf intr oc noMamjv. ro.y,Haipmep,npa8eatrrtrnpaBerhd roranrcyjeMo dllr,.lld. v6yayne, r(o diieapyDqfie HarnaueHo, kaaa ro8optr oonaparefHrM ABe npaBe.y napanenHe a{o Herajy

dlaj e! oj&B

Ase o"F.oFF"!dr ipa.a Harhsaso h{Mof

r'

d

d'd {'f,p Llajy 4

cldl

ojpoB

4

iarHM ipasaMa

rac(yA I npaBy, nocrojr raqsojeaHa npa0a, koja caapRr rac ry,4 rfopM M le Ha npaBya 3a

jeraqka,a,ipe.ek HopMarc tr I ipase je.re pa.roFEe oa racrc,4ao npaBe a. Ha upEHy

{" r,q)

.alpft

n od{'np,1"4" rac(y n

I

napaienRa ie c npaeoM d

d.

ayxHHalyft n'li


(

2 v )

a) 6)

@

r(, r pa aalo6rieu aBe npdBe kojece. ery 06 e*d ar o6se*,br{os nperer npadri

npe.a!r, npasoyrao{r nro nankD:l !a ao6rjed !se naparsHe npa!e. 06eneM ,x,

xo;.

*

o.ery, "p*uy o*"n

ha ha

a Koie

ct napar

qNqr naparq(e ca r!ruoM,4Al qhqr Mho*ralke ca trsruoMral

nparrx (oje caapxe raery npdix rcje .aapxe Ewy f,perx Koje GApxe ucry

OApeieHocr paBHr TaqHo jesHa

. rpkMa

p*aH oapeieHa je:

HercrueapHh raqMMa

. jeaHoM npaloM

He?

I Dq{oM {oja jojHe nprGAa

i

13,

oi-----l-.-


. a0eMa npasaMa {0le ce ce{y

. aoeMa napanenfm npaBara

A'o aBerac€ieaHe npaBe nprnarajy pasH,, oMa f.rac racka re npaDe nprnala paBtrtr, oaRo.Ro npaBa nprnaaa pasHr

| 3 np.Be p nprnaaajy pa8Hf d, uro rHrcrna ia\ra c npase rra obe nprnala

Taqke A

NalDrarceBopoyrao,4SCDy paDHr d Aa ntr raqG 0 npe.e{a !,larosana iptrn.!a roiDaBHtrr o6ere*r ca Ex Fjou ase iaq(e koje,ypHonptrmaajya ojpdHH.

npaRyrcu(e oiojnptrnaaaryraq(e,4/D,D1 crpaHy koq{e kojojnpfiaaajy npase crDr I DDr. leopa3flh (ojoinptrnalajynpaeans,racKacl reo paBHr kojojnprnaaajyracke A, c tr c,.

Halprai y Ge.ur (Balap,43CD4 j3,C jDl d)oleHq,npaHy Gaaparcjoinprnaaajyraqrc 6)o.eflcf .rpaHykBaapa {ojojnptrnararyipase 0)oceHcr crpaHy kBaapa rciojnpinaaajy npdBa r)o.efcf aeo paDHf rcjojnDrnaaajyraqrc 3, D


Kontr(o paBHtr oapel)yjyreMef a,4,

To

.y pa*rtr oapebeHeraqra(a 3, C 6) ,4,3,

a),{

t' (-,.

korf,o pae tr oapebyiyfMrHa A,3,

('l

(orr (o HarBtrue paoHr onpebyjy!Be tr!3erac(eM' rer npa3,x?

llpoBep'l ura

3HaLU

1.lGirrc paBHr oapebyjy ieMeHa,4, 2. l(on{ko je pa6Hr olpeieHoa6eMa

MrMOhtAlHl,i

FPABUI.l

Y nptrponr ceno Rara3ho nO,MepeaBa npaBqa koja ce y{purary, a HeMajy rajelHrckri

raqa{a, kao Ha 0trqr.160r bei6erHoc y .aobpahaiy, Kan roa jero Moryhe, rrpaiyjy.e nyr$r yasa Mr erueHHDoa 3ara{Benpa8ue {axe oAa ce MftoMare. A(o raMicnrMo Aa je o. (i ra@a npaBaq trpe!.raB^eH npaBos, A06trhedo MrMorrarHe npa0e. Taoe npaBe BeMajy

rFaHrq0x

nsc I Ne iprnaaajy

C4G npaBaca no jelHoM ar rc( orpeiyie le!fly paM3


npaBa H paBaH .

nph.alro-

. n:panerNo.r

nPase PaBH! nP6Be

r

paBHr

Tac(e,4 | 3 He nprnaaajy palHh a, a rac€ c, D I E nprnaaajy pa8Hf oje npa8e oapeieHer*M raqkaria iptrnalajy pasHr or Koje npaE o!peb*e rff raqkam He nptrnaaajy pasHr a?

o

-/ C6erac(e npaBe Mory aa nprnalajy pa6Hr I rala GxeMo !a npaBa Ha

npfrep, npaeapnpBnaAa pasHtra.

npa6a

I

paBaH Mory

aa.e sdby yrareoM nonoxajy aa

Fa nptuop, npa0a p tr paBaN

o

Ha

HeMajy

orqtr BeMajyijeaHfqsr

3d npaByp roia nprnaaa palHf dMr o psHtr o se a ijeaHfcor raca'akaaedoaa jenapanefhacapasHrd,lroranf.yjeMo pId

npa8a h paBaH Moryaa hajy cado jelHy rajeaHtrc ryraqkv Taad kxeMo aa npsa ip4eua paeH.3aieaRtrqka raeka npaBe H pa8Htr

,e.re ipecek npase

I

pa3hr.

af ipaBa p Ha cr{$ fMajy raieaHNKyraq{y P. Lseracke npaBe p, cer raq{e P, He npriaaajy pasHtr d Tac{a Pje rac G ipe.eka npaBep I pasdi d knr racG P ie npoaopHa racKa Ha nprMep, paBah

v6y!yne, a(o Rtrjeapyraetrje H naoeHoi cnarpahe tr paBar iaoarerHearc Hesalv sjeaHaqrx Eak.

o

aa.y

npaBa

59


2

Hrlprr parJtr0 rr'!e{€d 3E a npaoy, rora G,p*f i.cry 3 |

I

b) npa!y. (0Ja.aapah racry d Bl npdByd,oja G!pxr racky 3 tr

(oie iof ue (ourel3aDA,3,C,Dl

.e(y pao,r o!peiefly rac cra ,4, 0apenr sirooe npe.ecHe racre

ffi

To.y'srqeAB

Da, A,3. h D,C,. npe.eqRe raq{e.y, peao A, D,

(oje.yforue (ou(e

napareiHe ca

olpeieHorrackMaA, 3

To.yrBf ueA,3,,

tr

pa#tr

c,

2-Z

V)

D,C,, 4,D, tr 3,C,

fa.ntrq,

i-,)

je koq(a aBCtEFcH (oje r) trBhue (oure.e{y pasH oapebery rac GMa 3, c Orpeatr Brxooe npe.ecHe rac re. 6) Kojc.yr0r(e (oqke napanenre.a paBHtr orpcicHor

(.)

y.oc.ur (o(ky nBCDEFCH Mo Ha.ntruH ynpeuoaroM raa ry. Hauprajq o6enexr ror rp' raq(e koq(e koie.trrypHo np,naaajy paBHr Haupral

'tos/

1 0o 0p, 6'

1,p'r

A(o je pa@R oapebeHa

t!uda

Po!oovpoo,

llpoBep'1 ura 3Hau raq{e,4 ea, c ed,3 qa a) HaLrprajnpa,yb rcja caapRr racKe 3 | A. Aa ntr npa8a nphaaa paetr? !artr npa8d.ece paBaH? 6)HaqFajnpaBy. {oja.aapxtr raqrya I C. !antr ipaBa cece pasaHT Aa trr npasa nphnaaa paBHf? B) Hauprajnpdvd koja Glpxrracky 3 fl napaneiHa je.a paBHr d. 2. Haqpraj koury,43CDA,3,C,Dl a)l(oje trstruere(ou{eceky paBah oapeieHyrac(aMa A,3 tr 3j? olpeatr Rrxo6e npececHe rae{e 6)Koje cy rerqere rcure iapanenHe ca pa,Htr oapeiesoM raq(aMa,4, 3 | 3j? 1. Haupraipa8ab

dr


flpdBa HopMarHa Ha PaBaH . nPaBa HopMinHa Ha paBaH

. PadoFe oa rqke . vrao rcMetJv npaBe

!o PsHk r paBHr

o{ur

je npika!aHaeo ycroHiue. a) r(onrM je yrao nMeiy npaehx d h d? 6) Korrktr je vrao Ereiy n0a8u dr r? Ha

r) Koru| jeyEo BMeiyiparx dr r)

Kor

n pJ&r

tr

jevrao EMeby npaBu

c?

dr,?

rrol,Ir,ir,' rrl IirBl

nocMarpajMo (Banap 1RCDA,3,C,D,

Ha

cnrur

3naMora cy qdropoyrtrogi,4aCD, ABB11, BCCpj. npasoyraoHfutr. 03Hacrdo ca d psaN rcja cnp4r leMeHaA,3 tr C aaror {8a!pa.8ei csspro reMe,leMe D, npaooyraoHf(a ASCD nprnala paBHtr a. npasa , oApeieHa rac{aMa ,4 rAj o6paryje npaB yrao ca npaBoM r, rcla capxx rac{e,4 | 3, h npasoM r', rcia caapM racken tr D

KopK npee

a

Moaetr kBaapa

I npare.

r rr8par kontro

je yGo BMeby

'oiajeoapeieHarackaMaArc. 3a{^yqyjeMoaa jeoH npaB O6enexrMo ca Eraqq Ha r8trutr DC. Kaaa trrMeptrMo yco uMerry npa4 d , npse . koja je onpebeHa raqkaMa,4 tr A rrBpa|neMo aa je r oR npa8. BtrarMo aaje npsa d sopr {aqa@akynpaBypaBH,d{oja

caapx, seHy npececHyracryA ca roM parHi. v ror dyeajy GXeMo aaje npaBa 4 HopMantsa Ha paEaH d

61


no(darpajrc nparyb.oHa (a npaBoM a I npaloM /o6paryje npaBey oBe. Ha Nfl HacH MoxeMoa:yrEp Moaa jenpasa,rcpMarha Ha paraH koja je oAoebeHa racraMa A, D , /4. (Mr noaBaMa d{dl.

cnr$oroMe, ra ocHo8y rcra uro je npaaa d rcpMai|a Ha npase 4 , , sabyeyjeMo aa je sopManHa Ha p.HH rcja ie orpeieHa

HaqFaj pasaH o r raqrc A eo,8 eo r Ced Haqp.aj ipaly t {oja

j.

HopMaHa Ha pdaH

r6eHy npecaHy taqry ca paBM d a) KoiAt je yEo B{eby npaBu 6) KoarKr je y.ao ,rMeby nDaBu

!F

npale HopManHe ru

ooseM

, k 18'r t I c3'?

d ca

,'

jefty

dnb


3auro je rBrLra koLr(e 3,C, Hop{arHa Ha paBaH,433,? 3aro urojeAya Sjar HopMarHa koje nPrna!ajy paDHf /, R,3,.

Halyft,4,R,

tr

33,

(oje .y trofue {Baapa ARCDEFCH

;__l' nmmjf rqHojeaM npda (oja. 0aBHrr (oja je HoDMania HarvpasaH HanptrMep, raraq{y/q tr pa8aH a nooojtr raqHo jeaHa npa3a, Koja caapa, rac{y,4 | HopManHa ie Ra paBaH o PadojaEe tr,Meiyrac,e,4 | 0as|r d ieoe paaolase oaraq€Ano raq(er'rcja ceaoarja co npedqm raqka HopMa4e 13 raq{e,4 Ha ry pasaH

DqG,4 €D I rcM 3 €p. HeG npasa {oja je NopMaiHa sa | rcja.aapxtr rac ry 3 cere PyDqo R' arc je padojabe o! Dcre 3 ao paBHr PjeaHaro 12.m, a pa.rojaEe GMaiyraea{a,4 | RreaRako 16 .m, rorBo je pa.iojase rr ebyraca€,4H3'7

Ade.y psaH

8,

paBaH F

vraL

f:

rcrr rp.6r

r

He{a npa6ap.ece paBai d y rac{r P Oaaoepr onpotr o^Hyra{ry,4 fla np*ojp nooojtrEeHojeaHa npda rcjajeHopsanM H, peaHd

r(oja.aapxtrrac(y,4 Hehieronpa8a,.oHaipeceuapaBaHoyrdc$,4'

I olpeiyjy raqHo jeaHy npaByp je.reyraotrrnetynpasepIpaBH,o. Vraoksdeiynpa,trxprp Tac'e P tr,4'np|iaaajy pas8, c


Naupraj rcuky ABcDr,3,c,D,.

a)HauFaj{ obenexr 6)Nauprajr o6oextr a)

Ayx/4Aj

ca a yrao r3uety npaoe BAj ca 6yrao trrMeiy npaoe,4,c

I

r

HopManHa je Ha paoaR,43c.

Yrao h3Meby,43

r 34.

iecre yraoe.

6)Ayx,AtlHopMarHa je Na pa!a6 Y.ao BMeby A,C tr,4cjecre yrao^BC. 6

Haupraj Kouky,4ACDA,8,C,D,. Haupiaj EMeDy npa8e AD, 0 paoHr:

{ o6eneM y.ao

Haqpraj rcury,4RCDA,A1Cr D,. Haupiaj

I

BMeiy

npaBe 3D,

{

paBH":

rq!,iMcr_9n

oEenexr yrdo


\

lBe

paBHh

Kojf je i|A napa.e aic vaoM o6ojeriM y iraso? Kojr cy oa o6ojeHu 1r,rod HopMa^M Ha xtri 3{a? Kojr cy o,r ooojeHux 3|Aoe HopMan, Ha noa? Kotu cy oa o6ojer{x 3iaora sopManHh D nra+oH?

ABe paoHr Mory

.e Hahr y seaehM nonoxajMa

ABe paBBr Mory aa rMajyrajeftNky npaBy.Taaa raRe o!a ce DasHr.e{v. t6trxoB npecekje npa8a.

Ha nprMeD, DaBHtr d

I

F nptrkaraHe Ha

lprexy

ftajy

65.


HeMajy 0

3a paDHtr roje ce PaBHf

yf 6.y

r

rjeaH qdx

B{b.

o seMalv iaje!tsrqkrx raeaM.

nouaiajyMtr

ndpaneiHe,

rro

HeMajy

3ajearrcox raca{a (aHeMo aa.y naparetrHe.

ranrcyjeMo:

y

6.

pa8|r

0

|

A cy

iapaien{e,

v6yryne, a{o Hrjelpyraqtrje HarnaueHo, Kaaa ro!oprMo o napanetrHrM pasH,Ma, MrcnrMo cMo Hd pdHr koje HeMajv eFaHrcornqaG.

Akoffe paotrr rMajy 3ajeaHrqky raqry, oHAa Najy | 3ar'e!tsnc{y npa3y.

paMH

c

no npaBoj,

tr,llp

Aar je rBalap,43CoA,3jCjoj. y (areoM.y nMoxajy: a)palaH oapebeHa rdcuMa ,4, D tr D, pasaH oapebeHarac{aMa D, ctr cj 6) paBaH o!pel)eHa rac{asa A, 3 r C i paBaH oapeieHa raq{aMa,4,,3, I c,?

qr...........-.-1.-...

:-------------l

3,


aatie kBa!ap,4RaDAj3,CjDj v

I

a) pa8aHr33, paBaH DCC, 6) pasaH,4RC tr paBaN,4,DD,?

I ntiliits., r

1PP

f, rP

!f

ABe paBHtr.y RopMarHe aro jelHa oa

npa6y HopMansy

ra!pyry

Ha nptrMep, pasHr

d

tr

brx caapHr

paBar.

p cy sopMMHejep

.aaDHtr nDa8y, rcra je HopM

p

Ha Ha paBaH

Yr.., rrMchy,qee faBNr H*a jep ipereqs ipad pahh

dr

aRp

a

B

Koja nprMAa npaojp. olpeahMo npor3so^HyraqKy ^4 HauprajMo y paBHtr d npaDy a, HopdarHy Ha npa8yp y raq{r M, tr HalprajMo y pasHr p npaByb, HopManHy Ha npaDyp yrae{h /t1 vrao Ddeiy npaBu a tr, M MMo wroM BMenv oaer otr I.

ABe

pas

cy {opMai|e a(o je yrao

Ha iptrMep, pa8hr d tr Bcy HopMa,He jepje y.ao hiMeiy npaBtrr d tr b npaB.


nprMep, Meiyco6Hor a)

rpr

Tp{ pa8Hr rMajy sjeaHkcry Dcky, racKa,a je .ajeaH{ek racka pa!H{ c',

6) Tpk

np

r)Tpr

0

oAHoca

Bsr.

pau{ rMajy sje,lsrc{y npdy. a

dje

sjeftNM

paBH, cy

paBH,r

npea pasrcC F

ir

napeehe

JeaHa paBaH cece ase

napand{e pasHr. PaFtslcece napatruHe palr| dr F. npeceqHe npaBe cynaparetrHe,

aIr.

A)Tph pa4r Meiycob|o e ce(y u(o Aa np44|e npaF napandHe d[ I..

'

cyh{xoF


OproroHa,rHa npojexqnja raqKe H aVXh Ha paBaH

6)

I

il norots,r.iHa npa crirLfr. r.!

(e

I ntRr

Ha n!.rq)r

rac(a,4 6ah be HaupraiMo npaByr (oja 6!p$ raqky.a I HopMafta je Ha npasyp npe.ek npaBtrrr H p orHacfto ca,4' 3a racry,4'kaReMo aa je oproroHanHa (HopMdM ) npoje(lrfja Dqre,a Ha ipasyp. opro.oRanHa npojeqrja Ecre Ha np yyE'jeEcu

Aare.y

npaBa p h

Haupral

ayx,43 {oja He npfnaaa npaBol p. (poi.Baryraqky laMtrcirlona.Mo avxtrAB Rau0rantr npa8e HopranHe Ha npasyp tr tra raj@qsts oape,4Mr oprooHanHe npoje(Lriieraca€ ayxtr AB Ha npaByp. Ha ipa8ojprobfneMoaya x s AyMAs HartrBaMo opromHa,HoM npoje(LrtrjoM ayxtr,43 Ha n0aBy p o npdByp tr

oproroHanHa npoje

{rrja nyHi

rcieipad @jojip,naaaaya

da

lary

npasy

aje.rracM

RopMaiHa Ha npa8ya.

y cBarcMrpy,oM cryeajyoororoHarHa npojekqtrja ayxx AyRtrHa oprorcHdHe npoje(LrHielareayxtr

Ha

lary

Ha

npa8y HHre Beha

oalyxfHe

ndrd^ui oie!v6n'palq r lprBo{-drot Plpopnpo.dr-!vF,Fr npoje{qhje lyxf je!Ba(a rvxtrHH lvHn. a

69


Hauprajy oec$ npa8yp oprorcr&He npojekurje

I

racrc

Rao Ha

! frDcau

o{ur. olpefi | oEs€xtr B

HauprajyoecqHnparypInyMkoFao|qroapeaHIo6e oprorcmnHenpojekurjer fx ayH"

Ha

OpToroHanFd npoj€(urla raq (e

aary npaEy,

I Ayxf

HeM rac{a A He trphaaaaarojpasHr

r. cnhcHo rca oapebmaaa oprorcts He npojeku"je Ha npasyi HauPraneMo npa6ys koja caapx" racryA | {oja je HopMaitsa u paBaHr. Mo

tr

no(aran .Mo y npdxoA8rM tre{uhjaMa aa nodojr 6Mo jerra npad rcja GADxf aary Dcky | {oh je ropMMHa Ha aary pmaH. npaBa s npeceua paBaH ryraqo,4'. TaqG,4' j. oprorcw{a npohqrja ra{re t Ha paBaH ( Padoiabe oa raqrc t ao paEiir

Hek ayx,4A Fe npmaaaaarojpau{ r. 3arrooMoAa cMo {pB cBaky racKy lyaN.48 hauFain npaByHopnarHy Ha pa8ak r. npeceqHe rac(e rM npasn r parH{ rjecyoproErarHe npojekurje oarooapajriu Eqarc A eayxr. 0ForoHatrHe npojerurje c!NXraqatra ayx{ rB ra paBaH np,naaajy ayar r'3" {ojy Ha3rcano oproroH Ha npoje(q|jaayMrs Ha paBaHr.

Ha p.BaH

ea|


opro.oBarHanpore(ufjaayahkaaaryp&aHre.reracaa{oienpasa(ojojnptrna!aranyx sopMdtrHa Haaary p.raH y.BakoMApyroscrycajynpojek'r,jaayx,Hapa!aHre.reayx. rcjy npoje{ryjeMo H,BaMo npaj.Kquix1 p$aa. A|aaqa oontanaue d0a6" ieB-5oo!!. ,qdDop.,ir a'oFa, dpd c-p. jeryH4Ha pastr, jearaka . npore{Lrhi.(o oHAa npo,e{Lrire!y*f ayx,Hr lyHtr. PaBaH Ha

nalap ARI:DEFAH (ojajera, a) G oprorcH AaI )e

Ha npojekuriaracrcA Ha pa8aH 8cc? 6) (ojd jelyx oproroHarHa npor'e(!,ja aya,,4D ia paBar aa6? Bl UlraieopioroHdHa npor'e(rjaryaf ,]c sa paBaH 3C6l f)lxE je oprmo$nHa npojekqtrjalyxr aC Ha paBaH 3c6? a) urra ie oproroHanHa npoje{utrra !yR, Ac Ha pa0aB ABcl

a)OproroHatrHa npoje((trja rac

{e/

Ha

pasH Bcc ieoerac € 3 rro Mo ayh {3 HopMansa HaryprsaH

ie

_

6)oproroHafHa npoierliia ayatr AD Ra pa6aH 3C6je.re ayx 6C raro uro .y

!yx{,44 |

DC HopMarHe Ha

iy

paoaH

o)0prororanRa npojekufj, ayr0 rc ra pasaH 8CC jecrerac€ 0 raro uroje lyx l'lrl HopManHa Ha ry DaBaH

r) oproroHarRa npoje(uf ja ay$,ac Ha paBaH SCClecre 3C raro uro jeayx,43 HopMafHa Ha ry pa8aH, a

lyr

raq(a a np,naaaroj PasHr

A) OproroHanHa npoleruhla AyHtr,4C Ha pa3ats,43C jeneaya,4C lrro uro je ayx CC HopMarHa Ha ry pa8as,

71


Haupraj kolkyrlsCOAjsrcrDr. Koja Koja je Koja je

j e racKa

0proroNanHa

npoie'u,ia

ayx oFo.oHanHa npojekunja iBrqe AAj Ha pa8aH sCCrBj? ay\ opromHans npojeKqfja ayxtr,4CHa palaB CCjDTD?

He(aje 3'oplooHaiHa npoje{qfja raqke R Ha rasaH r. HipacyHajpasojaBe oaraqrcA ao raq(e 3 arcrac{a A npHnaaa palHr h aro je 33 = 6 cfr, a,4'a = 3 cm.

Iyx,43, crjd jeayxrHa jerHaM 12 cm, npoie{ie Heu je,4A =6cm,aa3 =4cm. ,,pd. rHai4 3 drcBd\iAd.yracle4,3

ce Ha pasah

:

c,(!'! eLrDdPpdBFr

6

"

od,r

r

(roaFa p"o

a i

r

:

a)HauprajMoraq(ycHaayxtr,4,4 rakora je3Ct A3.Ao6trjaMo npmeihdo f rraroptrHy ieopeMy Ha ipaBoymtr rDoyrao,4CR: R() = ABI A(' ha{ojerc=AA ss = zcm,ao6tria

o af-.\

6)HeG

ce

(pd,6eracke av*r aB Hanare

i:i:::ii:ii:ii,'i':ii:i" loko je,cA = 3DI RD=AD+RA'=10cm,

72

irli

\,

t\

I

"lul:*:

i.i....


\

bi

oproroHatrH, npoje(utrjaayk,,43

pa8rr rreoe lyR A 3' = 7.n, A 3'= 16.m

Ha

5.m,36 ra(..errcre,4tr3Han'e 6l(afdeapaHcpassaa r).a parB,rcrprHa paBHtr r fspacyHaj,cB arcie A,4'=

TlpoBeph tura 3Hau 1.

rac{e 3 tr c.a rcre crpaHe npale. uaupraj I rac(v D 'a lpyrc crpaHe npase d y oaHo.y Haraq{y s Haupraj oproroHartse ipojekqfje lvM 3C,3D r CD Ha nPaBY a

HauFaj npaBy

a

I

r I iac{e A h 3 ca f.re.rpaHe paBHf. capvre dpase paE{/ EidLprdJ.dr.ec / D .dLprdJoprororqnde nooe uv p Ayfrr A3,30 tr CO $ PaBaHa. 3. oForoHanHa dpoje(utrja ayxh AB da pa8aH trjecreayxt'3'. Ako eAa 43 v b,3A Tacke,4 r 3 Hatrare.eca hcre.rpaHe paBsr t. Haupraj.(trLry.

2. HaLrprajpaBaN

I'r A3-ll(n

-lr'n.qpd

npr{arr@se npooopHtrr o6jerara y paoHr Harrda.e npoje(oMheM PaeHtsa Kojv npojeqjeMo Ha:rBa.e npoje(qrjcoM paMtr, a npde kojfta ipoje(ryjeMo racrc o6iera ffi padH sM@Mo ipojekutrj.oM rpa Lr&a. Parn{(yjeMo qerrpanHo tr napanerHo npojekroBabe (oa ueHrpanHor npojemo6Ba @f ipojek,rrj.o + .o0 rdpdrP iFor 1po " rosdM'poFr ri o),id{edroreF'opnpoe

!, ,po.4 \

'o@@ 1"p"1e_ . lo.roir.dp"-1{o o.onpoP rosdn"f opoolr'.o poi.oesP

c

^,)]>

73


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