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i
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69
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y
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190
T
x= X
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36
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9
X
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n
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m
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= 12
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~-
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t:!.
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liHuihru
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I 2
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14
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i
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2
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nn. 5.
2
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2
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l. How did the bobY gof stung?
"[4,
46
58
65
70
75
48
59
66
71
7'15
51
59
I ,l
72
79
52
60
66
72
80
54
62
67
73
82
55
63
68
73
83
55
64
68
73
84
56
65
69
74
89
I!
I
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0~t_;~{f~a::G~ t~t;"&:e~~Y after 1he boby got stung?
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80- 85 s tl 85 - 90 'i 45
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3
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5
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9
6
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8
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8
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0.84
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1.03
0.87
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0.94
0.94
0.68
0.97
0.71
0.80
0.73
0.86
0.92
0.65
0.89
0.97
0.92
0.85
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1.08
1.14
1.00
1.05
0.95
1.01
0.91
0.75
0.68
1.19
0.73
0.85
0.66
0.87
0.81
0.77
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46
42
35
28
58
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33
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40
48
61
44
31
39
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22
55
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32
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41
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: t'i:HlJl g~1_,tH~Wl ~ f.fh~ t,?i ~ Al M I
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f•.Hl4osW ffl'i~~on.nrsr~~rutl:l ruu~tsit~""il t U
.A "" ""' 0 Q~UHAlM ABC ~~ AC bill~s~ 9
l
1
L.B ~~ AB bill~S~ L.C
ru AC>AB
t5irlmrh
'1
B
~~
1
L.B>L.C
tsitrn'lt:l~ AC tf:liUMts D ~tl5 lJ I 'W
ru AD= AB
'1
A
1
ttlQi 2
AB = AD ts-Jg MBD
A~ru A il~ LB 2 ~t~
=
dl1_,?itm M
fd~Qi~
L.D2
dl~t1_,nltS MCD ttlQiS
L.D 2
=
L.B 1 + L.C Gfl~ L.D 2 > L.C
thtu L.B 2
=
L.D 2 &s-Jg L.B 2 >L.C
&9J~ L.B
=
(1)
L.B 1 + L.B 2 tSH L.B > L.B 2 (2)
} Gll~
L.B>L.C
&r:J9l mm s1g ~unw21~ tim~ '1 ~ Li n u l.j
: Q~t,?itm
ru
1
d51d~ rh@su1 2 ffl ~ttl rubill~a~'id~ iJ~ fiHS d~@su~Hfl ttH ttfi tu~tl:l blibill'l§~~ lJ q I l.j I q I I
I
0
0
0
~~Gdl~ ct19Gctl~ 1 ®
'iuHu~n1 m tu~ri~@sus ffl'id~tl:l rum5~s~~il~rius gfhn sus ffl t'tu ttfiru'id~ Li ~ 1 1ql) 1 jq Li 1
Cll
eJ
0
0
1
0
ttl blibill'l§S~9Gt:il~ctl~~Gctl~ '1 I
n~ MBC'b'lS LA= 6x+4° , L.B q 1
dJu~~ ~s MBC ffl~ 1
58
thtH1~
= 7x+12°
~~ LC
= 6x-7°
'1
..
rufJw: nb MBC t!'im s
""
l
LA+L.B+L.C = 180째 (drutin~nbls MBC) If l<'l 1
L.C = 6x-7째 = 6(9째)-r = 47째
~t.H!H lJbffl~rntnti
~:t:llru
LB>LA>LC &Sl8&!'ili1Sl AC>BC>AB
W1~ti~ AC>BC>AB
<J
I
B
1Vifufl: nbm21bMtS8tt:i2~ tn .. If G'l tJ
l..i
1
AB = AC =AD
of
Sb
L.ABC<LBAC
""
2U1'i.fi1M: ABC thl!it1'1HWl~ru '1 t!'iti~ru
L~b sc 2Jms eM &'L:Jru CM AB
Sb
= CA '1
A
~HLti]mtsJti
BC+AC <J
..
nb MAMtf!i(flS L.BAM
= L.BAC+L.CAM
B
1
sl2J L.BAM> L.CAM ""
0
MCMYlS CM = CA tSl8 MCM dlHHnH1JlM~li151n~ru C '1 tf!i91ffi(flS L.CAM = L.CMA ....., t'Ul~&ag
L.BAM> L.CMA 'Y_ L.BAM> L.BMA
Qb MAMYlS L.BAM>L.BMA sl2J BM>AB I
t~::llru
BM = BC+CM = BC+AC (&Lmg AC = CM)
~ti&S3 If
AC+BC>AB <J
159
Wifmff:ft: 'itl'tmhn sยงHrillti s firuntit94 tu~ ttl-e1m8 n'ifitm nn-eru tu~'i::llth~m II l.Jc>~; U 1.1 I eJ n lj 1.1 lJ &
u
l:H15-efig tuStn'eli5ntStl 1 "' 1.1 n 1.1
'l
t~tl ~sfiruntitntl'i::ll S 2m, 4.5m, 5.8m
IN
1:fl SlJ ss
r:g n.
Sib
10.2m '1
ftHlH1j~ ffim@ tut1 tu& ru5{5 fi run ti ttl-ejm~ nLfit mnn
tl ~-en~ f!JS~
2m, 4.5m
Sb
'l
~~nm@ tut1 tutisJ~ti~tltLm-e
5.8m
stl 10.2m tl. 2m' 5.8m sb 10.2m lli. 4.5m ' 5.8m sb 10.2m 2. 2m' 4.5m
t tl-ejfi ru1 tim 'i5 sJ s~ li5') tS'i5 sJ
9l blJ s ffi tiqy ML'i::llU~ til slJi t mrul~ li5 ttllj1 ~ u
1
hl1 n
sJ 9'1-l::lH tliot m n.
f~ b ~s~b 9'1-l::lti~:~b ~ti1 tu-em nlJit mnn
sb 5.8m
2m ' 4.5m
4.5 + 5.8 > 2
"'nntit9? ("'nn ) "'-
~ 5.8m
fi run tit ru59lbti ts gffimLli
2 + 4.5 > 5.8 "' b1
2. 2m ' 4.5m
sb
10.2m
2 +4.5 > 10.2 rinm9? (-@srin) 'l "'10.2m
tl.
2m' 5.8m
sb 10.2m
2+5.8> 10.2
2m
"'t151Y.t9? 10.2m
\
fi run tit ru59lt!ti ~: s g@s ffititLti ttl-ejm~51Lfitmn.n~rumst9 '1 160
ill. 4.5m, 5.8m ~b l0.2m
4.5+5.8>10.2 rinB_t9? (rln)
10.2m
4.5+ 10.2>5.8 ilnm9? (rln) "'-
5.8+ 10.2>4.5 nny_t9? (nn) '1
Luifuu: Firu~fim ~:~~~n5!Eh i
n. 12, 11 , 17
g,
G~~j~GfSj~
4.
1. u~ ri f fii.1Sil u~ ri <!)
0M
1
f!i. 4.7 ' 9' 4.1 'i
' 2' 3
G
fl A
l:HiJ{hd~ ~H S"J t 1.1
fS A
AHl.!J. 'i Hh~ AH n
&'if!i
!J.
4.2. iLu]uitaJuu~ri
n
ri
L~tfilM AHB rtlL~tfilMtnh1Jl~ H tf!'lms LH> LA
sl~ AB > BH
LH>LB
sl~ AB>AH
!:lmsg nh'i~bfilM u
"' l
l.i
f1H9Jnmlish~fnh
H
B
A
Hthr~sMt~hm 1
u
i
'1
61
" : tHlllts B "' !'lb C 1
A 6i BC '1
H
) '1
tt?]mtijti AB Sh AC A
A
c
c nh ~AHB 'BlS LABC dl~tifflts LABH "'
l
l
l.j
t?i'QlS LABC = 90°+LHAB
i::lts~:sg nh MBC 1J
dl~tmru I
LABC> LACE sl§1 AC>AB
"'I
u
B Sh C tsl!Uh:&lh H (mg 2)tt:!ithlilu C' 'tHiHsl&rntiS'l~ ~ 1J 1 !.! .:: HC' = HC '1 t?i'QlS AC = AC>AB
162
tt!ru
C
o
~tl~~~fiqgnfl~ ~~riqJn1:llS~rulnthn~~riqJnU
o
~n~~~fiqgnfl~ ~~fiqgnudl~~riqgnt'c:~rn1:lls~Mnfi '1
dfunl'ff!tr t?12J ABC dl~tfilrul~rutl::lru AB<AC
'1
'1
tsiQ~~tfilrultsg
fU
I
n~n5 AH S~i:'i:ll::lJlS AO '1 t1_Y]mtsJ'u AH, OA S~ AC '1 ~~fi AH dl
tJtgw: ""
n
rifn~
A
,U)ru AO dl~~riqgn t?l'Q'1SAO>AH
ttnru
(1)
AB<AC (N'i:l~n'i:l) ...; ...;
ttfl ru13 rLJlti H ij n t
g B S~
0
t?19lQ)U1S HO<HC s12J AO<AC
ffl'i:l ( 1 )S~ (2}(?1tf1S:
B L----i..:,J,----l...,....----..,:,Y.
(2)
AH<AO<AC '1 !!ABC '1
'i5-lrlmm AB+BC+AC>AD+BE+CF 1.11
"""
A
163
l.
MBC =. !::JJEF BC = (2x + 30)cm EF = (5x-90)cm
?'HlJl Sl B C '1 p
tLfilYtsgtf:i2J
2.
=
PR
Sb
LPST
=
LPTS
MQSStl MRT '1
M
3.
hf:i2J
MA = MO '1
MT = MB
mdl !:MTO=.!:MBA '1
"
T A
4.
i:fi2J
AB = BC
tclJu
Sb
MED
AD = CD '1
Stl
B
D
!:£ED '1
A AB
5.
II CD Stl
TR = TS '1
um mm U1
"
MBT=. !::JJCT '1
A
6.
fi2J
LEAD = LCAD
stl AD j_ BC
mm MBD=.MCD sb AB = Ac
"
'1
DiD
7.
A
t?i~ P rtH~MG'tHi515ntsitrnnsgu2lrimls LA I 1J "' .,;; I
.
"" mMLnl:fllrulM'l:Hfln ABC '1 0<:$
Uult15U{]Gl PB
=
PC '1
c
B
8.
A
ABC ctlLntfflrulhl'tHilnn~ru A ~~ ""
M sb
0
N ctlcrMGnrun rutSLJ=ih AB sb AC tUb ffi 1
G'1
1
ttfiru P mcrrt.nGLUMblts BN 1
"'
sh cM
'1
"
B
ummm MBC dlL~tfilrulhl'IHiln '1 l/1 -..:>
c 9.
t?i
§UAB
~
0
CD Sb AC = BD tl1Jm2lb fiJ1 '1 IJ IJ G'1
=
tLu]umJuL~tfil rul ABC
sh BDC
D
'1
A BA
MWm~Ablru B -..:> IJ
10. ABC ctlL~tffl M'l:§ru '1 fi'l:§ IJ S'l
c ffl rilll b bill'\:§ Ln~
D shE
1
m"" BD
=
sh
ffi'i umm l/1 -..:>
"'
CE &SUL.ABC = LACE '1 B
ll.
t?i~ HOJ..OB, DB.lOB OP = BE, HO = DB '1
fi. I::J!OE := t;.DBP
12.
nbm2lbw1r;sg r;r:;§1 "'I
u
AC = BD
it.l
G'1
Sh
LC = LD
Uirlmm BC =AD l/1 -..:>
Sb
90° '1
BCIIAB '1
c
0
3. mm
tfflMSl:l u
21~ 'Lffll:l M~ t M~~ ffll:l
mtH1ih91r~u ij
E
K
A
c
Scm 3cm
D
mgL~tffl n.ns~
14.
B
F
4cm
M
21~ 'Lffll:l M~ &Ml~~ ffll:l
mti~f~~t912...,
1
B
s
F
R
15. PQR dl~tfflM~
PR
PQ '1
=
ru QR> QP ~~
mm LP>LQ '1
"'
A 16. ABC dliJHfflMMl:lQ'Hin~ru A '1 ""
0
U.1ti~tl5~~ BC ~018 CD= AC '1 1
tiblmm LACD>LCAD '1 U'l
"'
17. ABCdlL~tfflM~
A
~AMdlt1:11:lJ1S
ru LAMB< LAMC '1 tiblmm LB>LC '1 U'l"'-"
""
0
8. ABC dl1JHfflMM1:101~n~ru A '1
tiSlfif!lMtUm~O IJ
&
"'
fflfitiSlfif!ltummn &
ij
"'
A 1Mti~~id~ BC L:i
c 'ini:l D L:i
'OinmffiAD+AB>CD 'i tn " '
66
L:i 1
t~um21~nn '1 ij
ij
"
B
'' '
LB fflflLdtl AC H'~~ D '1 H mthn.mrufntl l
"
'S D
.., ' till15S"H'i .:;
''
\
BC '1
'\
H
''
'' \
15trlmctl AD< CD '1 lll .....,
' ,/'
\ / ' , AW-----~''~--------~
A
~-
Oeb
Q)'-::1
0
1
UMID15tictldru15nn't:HtHntmts LlABC u n ~
~
~mtl
N
L15 t' Hm~'i:fl H1 ~15 ni tl '1
B 1:..---.,!-:J------___;::,.c
21. ABC dl~tfflM~tlJtf;jru AB
=
3 Stl AC
=
4 '1 ttli1ti.ri::UllrJUhfflM 15~£!)
l <BC<7 '1
0
GM13 M '1 l n. tt15]mt{J15 MBM stl WCM
~- LMIDtl~Gl AC;AB <AM<AC;AB '1
c ~-
Ql
0
tn13M13 D, C, E I
~
~
~n1ntldH1ltlt9? LJ Ill "' "'-
(15ITTlR : t'!M 21 LDCE) c.s0
A
24. ntl~~tffl M ABC fi@g152lfArlll tS LB Stl L.C c; L.:; ~ 1
t'l.HH111n~ I '1 ffl'i:§" I l.J ,Oc;rj;
¢V
1
~IDLfl.mstm~fl BC ~UliDfflfl15~fl AB Hi~ D stl AC Hi~ E
'1
Oeii
LMID15.0Gl DE= BD + CE '1 25.
ntlt~tfflM ABC nS:Sli2lfltllltS LA 'ilit\Hl BC 'in~r3n.nts D '1 152lfil3rutt:Jrut'lli5 , "".:; l l.i .I LJ l <Ill u
ttsmn ....,
C
ttniD1Jii515Stl152lfi AD fflfi152lfi BA 'inti E '1 15trlmm AE "'
"'
LJ
lll-..:1
=
AC '1
----~~------------~~--------------d' B C
LABM
Sb
LMBC '1 21 ~ d , d' ,;
27.
6?Jtltl LBCE
LBAC = 90° , LABC =
bt
90° '1
60°
=
Sb
A B '1 D
Sb
ri LBDA = LDAB Sb
LCAE =LACE '1
n.
?l!lf!Sl LBCA , LACE , LCAE , LDBA
Sh
c
LEAD '1
E
28. ABC rtl'i~lltfil!lfl'i:Hi5 '1 fiSgti2l~r12tS LB l.j
!.!
""
IUWQ)rl A
,;
I
168
DE= AB+AC '1
LC
uuu~bti~~ BC 'if~ D Sb E dJtl[!
n. MEC Stl !'i.ABD rtlt~tfil !lfl IU'i:Hf!Gi ~.
Sb
'l
'1
1.,t1fltm5~m
0
0
'
"
nrulf!UH.RtHSu~tfilrul
o u~ £!) ruQ nn: sb@ ~159 tstytr~59JLfil~ 1J
o um mrun9 nn: sb'i'9 f'6u 9 tSufl1 r; m nn t m nn lin lfl"'-'
cr~tmnndlrl'Ujtfilnn
'i u~&mnn
rul:flsi_Eib4 I
S~ t15']'i:fl Si_Eib 4 sb~ 4 &Ul ti5'i:fl Su~& fil rul &dJb WJfiBl S I
run rul :u tS"e~ t Olb Sg L:lu2lb 9
"'
1J
iJDti~
ABCD 0
'1
m"" ABCD 'i:flS
dluf!tmnnmru I 01
II
'i
tytrubiltm~ dlu~tmnnt'1:lrul:flsi_Eibm5~ 1J
I
uuu~ "" m'] '1
"" ABCD m
tn59JLfil~ 1J
I:RUl S [AB] e
II [CD] sb
II [BC]
a~tmnnfnt! mtyr;ru9JLfil~
'1
rul:fls~tnt!
u
~ru '1
ttl ABCD dltU&ru9JLfil13 sb
LA
IJ
n~----------~c
B
&Sm ABCD
dltif!&filrul I
~~S~ '1
ttl ABCD dltmru
AB = BC &S'HABCD
=
c
A
Sb
dlun&filrul!;hl 'i 1
""'
D
169
nH~
B
ctlDJ ~ruSJLfflt3 &'i::Hlil:fl s~~ riHtl'ci~~ s ~ s~ u
l:flS~fn~~ru
1
ABCD
'i
u
90° S~ AB
LA
ctlDJ&ru9JLfflt3 ttfiru
=
A
BC tSH ABCD ctlfilt~ 'i
D
- - - - - - /t._tJUiflfj'{fn1!
~
/
/
7
Lm ru 9J 1J11 '1:3 m11 ~ t mru1 rt~ rul:fl s Ltl ~ ru5'1:3 u
~Cl~'J 'i
1
ttl ABCD ctlLmru9JLfflt3
t~ms n~--------------~
u
II [CD] s~
II
'i
l}fgfir~g 1 : »~rl1J~S~ti5'J ttinDJ&rulULfflt3ctll!itmMria~s~ u
ABCD
mm !J.ADC =. !J.ABC
ctlDJ&b159JLfflt3tt!ru
-...:>
u
L~, m ru1 A De
sh !lAse l:f1 s
AC LA 4
=
0
'
LC 2 (t3~Hun~) 'i IGV ell 1
t!msg u
!J.ADC=.LlABC
'1
u
(nHUl
'1:3-d-
'1
&~LilS llADB:!lDBC &~Lil
2 : AD
=
(nHL.fi '6-d-
= DC
BC , AB
LA = LC , LD = LB
1mn : 151iHlli'6sh~rn5'6'&s trn9JLfil'6~s ~ 1
ABCD
'1
ll
dlL'iHill9JLfil'6ttlrn "
LA
sh
LB dl~n~ '1
"'
0
u
ummm U'l
LA+ LB =
1soo
'1
A
"
ML"l:f7 W'iJ
n;p !'# :
t~'Hls [AD]
II [BCJ msht:tH.l5~~ "'
l}tgug@ 3 : ~~~1Jh tsLmru9JLnl'6LunJ~~Ln~linf1unn!Jl rnt{Jh~ o tUTl u
u
mon~g '1 .!!
""1
ABCD
dlDHill9JLfil'6ftlru'!:fls»~~1Jh [AC] sh u
Uhlmm U'l
"
0 GliTMUnrunrut~Qb'&s 1
ML"l:f7 W'iJ
LUnJ~~Ln~ 0 "'
u
C]
G1
Sh
'1
n;p !'# :
Lfitmrul AoB sh
lloDc 'tfls
LOAB = LOCD
A
B
('thnfdnb) 1GV "' 1
AB =DC
(Ldhrn5'6Lutru9JUD'6) u
1
LOBA = LODC
(~ffihlnb) 1GV
~msg
u
AAOB:AODC
tf-1012
OA = OC <I
sb
OB = OD '1
~fitS g0 u
G1
1
(nHJl '6-d-'6)
o mnMunrun ruts o moMunrunruts 0
"
i
G1
1
G1
0
dliTMUnMrl rnt{Jh tS 1
"'
C]
"
[AC]
Sh
"I
'1
O~tfilllil ABCD 'inS AB = CD ShAD= BC '1 U~QJGl ABCD
dltytrt59JLfil'6 '1 u
fiiJhAC u
L~tmn.n ADC AB
=
sb ABC 'i::flS ~n'i:3) ....
CD
Ac Loha'i:3 1
!:Jm:::lg AADC:t!.ABC (~laM d-d-d) '1 u
lcnn: LA 1
=
LC3 m9mh~mn5nhtf.lms i<v ..
II
(1) '1
l
LA 4 = LC 2 m9mb~mMnbtf.ltnS jG\1 c;
II [BC] (2)
l
r11m~ s ~ il ~ 'J n:n gu~t m M ts 8ctli,YtilllilJLffli:3 '1 u
m'ibfflrul ABCD 'i::flS LA l
= LC
Sb LB
=
f!Jrfl
LD '1
ABCD ctli,Ytill9JLffli:3 '1 u
' '
tf.l'i::flS LA+LB+LC+LD
'1 (
=
ornun u
r;m M) LA
=
LC Sb LB
=
tf.ltnS 2LA+2LB LA+LB
= 1800
LD ( tu'i:3~n'i:3) .... ....
=
360°
Sb
sh LA+LD
""
LB
=
LD
= 1800
(rlilltin'i:3nbm2lh) u jc; !.1 0
I
II
II ~msg u~tffln.n ABCD
W'ILtitill9JLffli:3 '1 u
GSHAlM ABCD 'i::flS l
72
II
sb AB =
DC '1 ti1;fl£!JG1 ABCD ctlLtitill9JLffli:3 '1 u
9m
M['i:n l!J'iJ f[{l If
:
M~H~fl1Jh AC u
1_,fitnHl.!l f].ADC Bh !].ABC 't:llB AB =DC LACD = LBAC
AADC=:f].ABC (d-~-d)
tSlg
~tfln : LDAC
LBCA !\5msi9mh~m
=
"'
II
IN
sb [ABJ n[Dc]
II
tH'itnH1.fl ABCD 't:llS l
"~
tltH!H tHHffHlJl ABCD II l
7 : dhHHnH1.fl~ttl't:ll2Hhfl19bi'tinH1 ffiH'i~G n u n Li Li Li lJ9MfJ99 1
1J
Unrunru n
<lot
h
tSlg
U~tnH1.fHSgdl1JHru9JLnl~ '1 1J
~1JhU~tA'lMn
0 dlGn.f1UnruJHUG2
rudJb
II
ABCD dl1_,titru9JLA'l~ '1 1J
M['i:n l!J'iJ f[{l If
: A
1_,fitA'lM AOB Sh LWDC 't:llS OA = OC 0
â&#x20AC;¢
"
L.AOB = LCOD (~9run~
D 0
dlUMUnrunru&S 1 n
OD = OB
~tfln : L.CAB
ttfl ttl
"
'1
L.ACB = LCAD
II
'1
II
'1 tru9J1_,1ll~ '1 1J ~
0
fiJ'i.fTUif.'it: tl!l't:llSUntnlM ABEF Sh EFDC dl'i'i.H IJ
!
ABCD dlLmru9J1_,1ll1:1 '1 u
l2
C
II
~msg U~&A'lM ABCD I
l>?J
LACD
=
UntnlM ABCD 't:llS 1
0
B
Li
..
F
.----------------. E
b{j ABEF dl'U5WHilJl,iil'i:HS'U AB
=
FE ,
ij
II
(I)
t{j EFDC dlt,utru9Jl,iil1HSlg FE= DC , II
II
(2)
fl)'t§ ( ) 2b tf!'lQlS
(2)
II [DC]
AB =DC ,
~msg u~tfil M ABCD
<t
dl'U5tru9Jlfll't§
'1
II
"' tmru D
0 " dlun.nu~8ts
l
ru9Jl,iil't§
.
o.,
ru'i:fls9u1b\jru
'1
.
II ~
A
""1
A; - - - - - - - - - - ,B
t tn ru Wl~u~tm nn tnt! nmt,m rul{lt,Al't§ t Sl gl,f!'lti II 0
~
~
I
I
nn:9lbHtum tu'U5Ul59Jl,iil't§ndl ruQnn :m tuu~tm nn IJ
tSlg
LA = LC , LB = LD
11 D''-'----------:c
llli't§tStytn511Jlfll't§) IJ
t'lflru
LA =
=>LC
A,-,--------,8
=
_j
LA+L.B = 1 +L.B =
tlmsg IJ
=> L.B
=
LB = L.C = LD
900 t Sl g L.B = LD = 90o '1
DL----------'c
f)ffl
ML'i:fll111J f[fl ?#
:
t{j ABCD dltHWilM i
..
nh !:s.ADC
&lrJg AD
= BC
ru:rl§ hn:s 61 t mnn fnb ) A~----------.,~
ab !!.BDC 'tflS
I
AD= BC LC = LD = 90o
!!.ADC=.!!.BDC (n~
0-'1§-d) '1
~tnn: AC = BD '1
titH£H AC u
= BD '1
tf!i'HlS ABCD
ML'i:fll111J f[fl
dlLU&ru~Lffl'l§ttiru AC
= BD '1
tl~[gdl ABCD dlu'ttfflM
II
n:
&u ABCD dll}5tru9JLffl'l§tlrlg II
AD = BC (~hru5'1§tstmru~Lffl'l§) u
l
AC = BD ( DC
dl~hJ'I§ I
&~012 !!.ADC=.!!.BDC
~Oiii :
n~
tl-d-d) '1
AD = BC '1
LADC = LDCB
nJiru
LADC , LDCB m'l§dltitltS'I§ I
tlrJg LADC+ LDCB
"'
=
2LDCB = l80o si2JLDCB = 90° '1 l:H'HS!l ABCD u
t:htHHffln.fl l
'1
175
fil~ffiY M 2jma MR
'l5
Rffi MP+MQ
~
= BH '1
A
Qb1.f\tfilrul BMPffiS LPBM+LBMP = u
Qb1.f\tfilrul MQC ffiS LQMC+ LQCM
=
u ~
0
ttfltu LQCM
Q
lfOlntntfi1lJHuYQ1n ABC
= LMBP
f:jtJtS8 LOMC = LPMB 1J
-
ru
= LRMB
LQMC
~SU
= LRMB
LPMB
'1
1.f\t fil nil BMP SbtWitfil MBRM ffi S : MR = MP ( fUY~fiY)
'"*
LPMB
'"*
MB (~b~Y)
LRMB
=
l
llBMP:!J.BRM
(mnfi r:l-Y-0)
lmn : LPMB = LRMB
=
= MP
90° , MR
=
~Qlfi RQ = BH tn RQ = MR+
sg
'1
L.RQH
=
'1
MP+MQ = BH '1
: tf:i"tf!StJ£11 filrultfih ABCD SbHb~ n
tihl un
'1
g
-*
I
2.
LCAD = LBDA '1
~
nnt m nn t fij
m1m !.J
~'i9h SYtuC) ttJfifj£1 tfil nn' Mdl'i~tfil nn
'"* w 3. idhrumsb~ntiYtS11ntfilnnti6usm wl l l '"* l "" 4. Hh~'i9btS1Jntfiln.flt n w 1 !.J1J
1J
76
1.1
'1
dlnnt fil n.n fnb ttfi rul!l m u
, He
.1
= MP
u
'1
n
LQRB
=
900 dltJntfilrultfih '1 I
tf!l'i:!lS ABCD ct"J1!.Hru~un~ttlru AB =AD '1 tl~(gGl AB = BC =CD= DA '1 iJ
ML'f:fl UJfj rrJ7 fi
:
t{j ABCD thijl&ru~un~
B
l.l
tSH AB
CD Sb AD = BC (!Jlbru5~tS
=
I
c
A
D
tlmag u
AB = BC = CD = DA '1
ML1f'l UJfj rrJ7 fi
: B
dj ABCD thtHHffHlfl& I !:.ABO Sb !:.ADO 'i:fl a AB =AD OD = OB
D
AO (!Jlb~~) 1
si~
t:.ABO:=t:.ADO (Fi~
~Qln : LAOD
tmru
=
d-d-tl)
LAOB '1
LAOD , LAOB th~dlUtltS~ I
2LAOB
=
1800 , LAOB
"'
=
900 '1
tf!1'i:!l2 ABCD dlijltru9JLfil~ttl II
177
.
B
!iABO Sb !iADO 'tllS : OD
OB
=
LAOD
LAOB
=
D AO (~b~'i§) I
2J
!iABO: MDO nHW(d-'i§- d) '1
l(fln :
AB = AD '1
ABCD th1YU15~Lfll'i§ttlru'tlls AB =AD ffi'i§Stl5'i§SttHI!'i(flS u
ABCD thtHHfi1fl1HM '1 I ~
.. B
!iABO Sb AADO 'tll S OD
=
OB
LAOD
=
LAOB
AO (~b~'i§)
D
I
il2J AABO:!iADO l(fln : LBAO
=
n~
(d-'i§-d) '1
LDAO '1
ri 1J b '1 II
A
78
mm AOBJ th13WHfi1rult
-...:J
I
'1
1
m11 fl.fJ ti'\3 ru f tl ru
d5 AOBJ mnHru9JLn1l:H~Ol2
=
OB , IB
=
OB
ll
ABCD
I
tiltHHffiMtntl 1
t~OlS
AO =DO= OB = OC
t~OlS
AO =AI
~msg AOBJ mnHru~Lmattlru'tf!S ll
;ot~mti Ao =AI lJ I
mmHmMtM--& "~ 1
'i'i.J~fi~: ~MiSfUUl trl <II
LDAB
l..i
tHHffiMthl ABCD
LBDC
"!
~
I
Stl
c 2
rugn.n:mu mt~ tilti~tmn.nr~t'!:lru'tlls~&ntl'\1ru "~ rr:nrucr~&mn.n&~mnnru~Lma&su ll
mw~mLmru9JLma
"!
u
tilLmru~tJLma u
m&H:hu~tmMtnht!:lruffisiJlhmtir~~
..
rr:nruu~tmM
"!
1
tSl8n1UntilTIHru9JLma "!
~msgffic~ffis
ll
tMshmHmMtnh --& I
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3. '~Ohru.mmttutHnshrMm lJ c:;l.j c; ~q1
4.
" »ru~g
tatijl9itS1Jlh I
I
5. Hh~'i9htlSffi n lJ 1 "~
ffi'ini.1dn.ncrn lj
6.
I
<'I
u
mi:!Jl9atsiJlhru.mmiinJ~g
.
I
1
7. »h~HilhtilnSllUSl~m'ts~n.tla&Uirumiin;~g '1 n
lJ
u
w
,:;
1
tiw
1
1
l}igfffJYJ : Qh~tffi M»~~tti ru.lJIUU 1
1
ttflrutMshmRnrunru;ot~gti '1 »h~rsgctfTlm ""'
~
tmru;prutSLdh rhLn~uutiShiJlhgu
l,'; I
n
ll
nn
I
"~
79
0
,13
13 E nrunrn [ACJ '1 G1
II
'
uinmm lt1
"
DE= l
A
u~ru
~ms DE= EF
.
D
•
.. ..
0
M ~ [1U1~13
13n
t blililJ1J1l'ij
rn E
"'
.........
' ''
' ,'
B~--------------·=·c
lJ
I!
tEm AD= CF ,
sl~ DB= CF
ttntllAD =DB
II
136HffHl.ll BDFC "ddS DB= CF , 1
tmg DF
''
F '
= BC ,
I I ruDE= -DF =
II
2
1
II
"' ' DE= -BC 2
' '
II
"'
tf:'ifflf;l
13&'Hfi'HWl BDFC "ddS [AB] II [CF] ' [DF] II [BC] 1
~ffln: BD = CF"' DA
= DB (G"
13 D n
136'Hfflru1 DAFC "ddS DA 1
t?l01S H~~lJ~
=
rn CF ,
S~
lJ 0 " t:Jmsg E dl13ru113nrunrnts [A lJ G1 j
180
) !:Sl8 DA = CF "'
] II bliE"'
t~'Hl2 b.ABC fnhUi~ B BO =
~AC
c]
'1
um Ill
'1
ML'i:!7 UJ'iJ fJJ1 rf :
U~t1Hl:H:iJ12
2Jms
BO == OD '1
t~Ql2tHHfiHl11 ABCD ft~ru'HlSHh6h9hiUnHlffl Li
n
1
L'1~Gn.f1unn.JJ1 ru
IJ
Li
.o"'
o £hLUHl59JLfil'l§ '1 IJ
TIHru!ilJLfil'l§ ABCD ft~ru'l:fls
dluntfiln.n I
LABC =
IJ
1mn : BD o ....'5f
=
o
1JtgVfUI4 :
i12J
AC
~AC
BO =
c::;
QhL5itfil
'1
bn~:l~bu;~sbn &Sl::~~:filn.ntm::
t'l§1:ljlS&
1
l
IJ
m~tfil n.n &nb&1:! ru'l:fl s~bu;~sht'l§i:lJl sm IJ
I
l S BO = -AC '1 2
ML'i:!7 UJ'iJ fJJ1 rf : U~ti5t'l§i:lj12 [BO] 2Jms BO
tHHfilnfl 1
ABCD
=
OD '1 t~QlS
'1
ttnru
f1:lru'HlSHbn
&2'H BO + OD
=
1 BO = 2.AC
C+!Ac = BD 2
BD U
"'-
= BD '1 1_?tru9JLfil'l§ ABCD ft~ru'i:flS
AC
fllJh
IJ
AC
= BD dlu~&filrulfnb
'I
IJ
~m~a b.ABC fnhLn~ B '1
2.5.
runnn:mnmnn 9
I
u'1tfilnnmru dluntfilnn 1 c; a ~hLruu~&sgnmmmn Li 1
m '1 "
I
ru~bfh
I
'~db '1
ru'l:flS'idh LJ
n~sLruu~
l
untfilnnmrufr:Hli'Hl l
"'
mruru'I§Qln(m9 "'
tsiruh:nbts~hm
1 Htfi
IJ
'i:l'ru t~tUTlm~Ql'1tSuntfilnnmruru'i:l'Qln I
IJ
1
"'
1J
1
'1
un filnnmru l
&t~ru'l:fls~fnb'i:l'rutUTlmun&filn.nmru&nb I
A.-----,B
l
"'
"'
2) '1
c
tl'i
LD st~ Lc
ru
tunm
~m~ '1 l
. II
II
A
IJ
ru DA
=
B
,.---'\~Z~7
ti?:'iQlS EBCD dlttnilllijLf'O'i:j 'i ti?:'i'i:flS DE= CB
E
CB (1_Elb l
tffH1.1l UltllM'i:jQl~)
~ DA = DE 'i
!lADE dltli&nH1.1l n5'i:jtfl~ '1
1mn
:LA = LE
ttfiru
LE = LB (~L~~
) ~~
IJ
~ ABCD ttlru
..
A
DA = CB ('idtifi~tffHUlmtllM'i:jQl~) lJ I 01 I
'i:j LD = LC ('i:jQl~fi~tffHUl Q
l
I
n AC =DB
182
tH!lmttl AC lll
L~tffHlll ADc st~ !lDBc 'tfls
"
'i
l
'1
tllM'i:jQl~)
= DB 'i
"'
B
t~'i:flS ABCD tHHnHl.ll I
ruftHli'i:flSO'l~
II [AB] &tfi ru
n
i'Hl{fl &li hq_~ tl ij ~ Htni G1 01 fi'i:HiJU t S
tHi &ffl M
1
ummm [DCJ 11 Ul --.:1
mru '1 <I
a~
11
EF = !(AB +DC) '1 2 ML'ln wu fJJI
n:
ffi'lยง C ~MliSln CE ffl ~
II
Ui~ G L~hfflnn AGE a~ EDC 'i:flS rdn~) "'I
LGAE = LCDE
1
" ED (E dltiMtHHl.ffi&li 1 51
AE =
."
rnn~
LAEG = LDEC
lifin :
AG
= DC a~
GE
= EC
'1
'1
Q~l!itfflM CGB 'i:flS I
E mttnru:;nrunrula I 1"1
II l l I EF = -GB = -(GA +AB) =
2
2
16Mf!fJ94: n~n~tfflM L;n cs 1 I
II
+
ruannnnrunrnts'itltl 1 n lJ i
lJ~ ~msiHliu~rir~~ruftlrnLM'iHms~m~ '"~ II
&~'i:flS
ABCD tHHfflM I
'lJ~ [AD] ' [BC] S~H~rilJ~ II
&li'i:flS E , Fa~ G , H dlGMfinrunru I Gl '1 ti!rlmm E , Fa~ G , H ~~tsiUtlli IJl --.:1
83
. ~~Uitmn.n ADC ma I
" ruts
" G marup:sn
ts-12 [EG] d1Qln1H:$J~laU~tffirul ADC '1 [DC] II
U1(fl~
'1
a [DC] II ~Jl~teJs;
" s;~a'l~taafitffin.nmu:ns-H tJ I <t
a
m~
"" "" Lrirl"l&ru
11 [EF]
(3) '1
"' maatlH:tl E ,
.. Fa~ G , H ,e. un
IJ 0
GM1 G E ,
g
C::.
&rutlSlntn~ru '1 <I
N~ts~~tlยงlrl
a~
a~ H
nfJti19N~tSH~fllยงl~ AC a~ BD
I
'1
G
: Uitfnrul OAB N~Q'ln t'iH~JlS
rl LA
= LB)
G
A
dltU!:W
n~ l!.ADC , l!.BDC oe
'i:flS
I
AC
= BD H~rilยงI~Grl t ffi M n 1
!llNUQlri '1
II
CD AD =
sc
Ul~iaantmn.nmru I <t
0
&'1'1
I
~mas
llADC: ABDC '1
ttlQlS LDAC = LDBC LDAB = LABC t?l(fllEl LDAB-LDAC
=
m~ AHAB mU~tffirul
mu (1) a~ (2) t?l(fl a
LABC-LDBC m~ LCAB
n tSlUUf:.lJlS dltU mt~l:lJl
"ts
u:rhruumrifn~ru, <I
184
'1
U
O,H
tt'iQlS
IJ
" " AB '1 G dlGMGnrunru&a I ~
: ttl'i:flSGrltffiMmruNUQ'lri ABCD i <t 0
G e/
Q
Fa~ G , H
II
=
LDBA
"' " [AB] (2) '1 9US
B
0
•
nnmn
2.
n.
t,tn'lhrtyHt,~hfilM ABC '1 ttn~ij ?
2.
ttnthru.mtfitfilM i.j .... i.j
!!!.
tfi
16
&tnnH
11ABO:I1ADO ii(g ? ~
I 6
II
tS [AC]
&UlW
ffl~ I '1
timmt:n
? D
?
nnmu1mru I 6l
Qbt,utruf{lt,fil~ ABCD , J l
3.
ttnnH
ADO '1
[EF] t:h~tH%~rufl:lrufilri n
Ul"'-'
!.1
IE = IF '1
U:hW&l_.tlG~&filM EFGH
n. t!!l2J
EH
= IF Sb~ LEHI = LFIG '1
ti~(!JGlG~tfilM EFJH t:h,YUU9~t,fil'tf '1
2. t!!!2J [EH]
II [FI]
, EH = IF
Sb
FG = IF
ru~LfilY '1
t'dn~l:lbis~uwt9'ln .... i !.1 \J '1 nnns·umfJ~2'1§WCJ 6 i !.1
4. l.Y'ruf{}Lfil~ '1 II
5.
d5~~ru&snHru9JLm~ms
N"'....' (2x+ 20)
II
" " x '1 (x-50) '1 !!!MSlnt~ta 6.
N~l,Y' ruf{lt,fil~& f:ll ru £)[1 II
n.
LA = 60° , AB = 6cm , AD = 4cm
2. AD = 4cm , AB = 2.5cm , BD 7.
t!!ltflSUHruf{ll_.fil'tf
=
3.5cm '1
ABCD
II
[CD]
tUb~lJl~ A' sb
c
'1
u ri
11 ~tiH9Jnmriffl'l§ 0
mri
[BC]
Stl
l_.n~ B' sb D' '1 U~(!)Gltl~tfilM A'B'CD' m,ytru~ilJLfil'l§ '1 II
185
"'
9.
0
B
..,
t fi'H1Jl ABC 'tf1 S E Si:l D th tHlJH1 fi ClJTH1H S I lil
8.
tl='l'tflStH'jtfi'HlJltnh ABCE shTIHrn9J1§1i:1 BCDE
'1
ll
u
10.
ifl fD m 15i' nH1Jl A cD mL5' m nn N'i:Hi1 fl '1 Ill ABCD t~rn
tl='l'tflSUfltfilrul I
II
" r munnunnm I
lil
Sl AB
1='1 rul
2.
l='lrulSlflt~tS
" "
=
XY
fi.
x
12.6
Si:l
DC
=
3 '1
DC= 4x-7 , XY
=
2x + 1l
B
AB = 2x+1 '1
1 . tt'i'tflSUfltfilrulmlllN'i:1'Q1fl ABCD t~rn I
"'
sh LA = 67° .,
12.
?'inn
g'i '1
tt'i'tflS ABCD
L.CDE
ill AD= CE
L.CED
=
'1
ABCD UfltfilrultM '1 l ...
.1 [BC] '1 tll:rl mdl BE = DF '1 Ill .......
0
0
L.OCD tn'i:1tl t nm ffi '1 I
186
'-'"'
II
A
;:
C
~
D
o
rHl1lnt~Lni{JlTI5'ru9JLm'iยง
o
nnnnt~u;I{Jlu~'mfl.fl
0
'"
IJ 0
ru
'"
t j:'j m~u;:J nJlJ 131:1 n ttfl
m~ fi nn 21 &~ Ln 1{11
hl13ntm nn &nil ttl ru&Mtl ruj:'j nn ~nil am 115 l
I
'-'o
IJ
1361 tm ru1 &nil ti:l rut24 ;;n~
has gtnsm n~m 115 4 tJ nm 2~
1
I)
ttl'in9nts13ntmrultn~~:sgms~m ,; lJ i.l' l "
8tlnmttl ,;
'I D
"" Lm1:1mnH15
&S"J&fn~j:'jnflS"J
Lni{Jl lswcru9JLm1:19lR9h tlh~an IJ
... run sh ma~ n td'in9n tS13nt mnn &l.J i.l' ~
~
':)
':)
~
fn~ 'I
1
TI5till9JLnl1:1 ABCD uHHl"" IJ l eJ
If'.}
'?
0 Q
cy
13~tm fl.fl tni:l tth ruun 2 &61~ fi ru t~(Jirfn fl.fl (tnHtfl
druj:ifl.fltsHnnhd~mn2hn't1 1 l,j ยฃ'1 I)
'I
I
n~ I'Jtm ru 9JLfil't1th H ~ fi
ru ~ M m ()] tl fi ~ ru
IJ
t9ifnhsh~i:lru.J~ 1
ctfiru~hru.J~tsnmmmmn 'I l
c
.. dln~~Tiltrul{jLfilY ABCD
[AE] ,
tel rulJl
u
tf16'i [DC]
u
dl n~ ~Lm ru
ABCD
.
A
B
.1
llAFD S~ llBEC "!
dj
ru
ABCD
thTI5&
" LADF = LBCE (Y'i6iHil) l
uu
0$
LAFD = LBEC = b =DC= AB
t~Ln~G~tfilrultn~ ABEF
=
t~~~G~tfilrultn~
t~Gn~G~tfilru1£PID
t~~~Tiltrul{jLfilY
ABEF =
ABCD
u
t~:mt~~~TI5tru9JLfilY
bh
= t~~ijlfi~tfilrul
ABCD =
u
ABED+
t~~~~&filrul
ru ABED+ t~~ijl~tfilrul
t~Ln~G~tfilrul
AFD BEC
ABEF = bh
~mag "'t~~~TI5tru9JLfilY ABCD = bh '1 u
: tl:h ruH~E'ilJ~ s~1Ji:ITI5tru9JLfilY m~ nm s~tfil n.n tia~s [1 tmg l~1fijl llJ
" • MnMntl:hru s
=
l
l
llJ
bxh '1
2
: cutmrueJunY
D
fJlt~cru
llJ
ntl\'fsi:l ...
NSi:l ...
Scm
l~Lfi~TiltrueJunY u
l~~~L~tfilrul
188
3cm r;mg
s = 3 x 8 = 24cm 2
S
=
~(3x
;]·:···················...
'1
= 12cm
2
A '1
/
8------~
B
c
muntffttj91: ttl'i:flS ABDF th'UHbl59J
A
JOcm
B
II
Lfi1Ytl:1ru'i:flS AB
fi.
=
lOcm , BD = 7cm
" x '1 t:Hl.fl SHH'H
""
" "" ~. ~nt~U'i!ffl'[!Hfil
n.n AFG
ocrJw: IN
&tlQlS FDxAE
= DBxAC
(t~Lfi!ff!LtHru~LfilY ABDF)
lOxx = 7x9 X
= 7X9 =
63
" ""' ~. ~nt~Lfif{JllJHfilM AFG
dht1tltun
t~1Jif{Jlt!hmn.n AGF u~sl:l 3.15cm '1 2
FE= FD-ED
t~LnijliJhmn.n
t~Ln!ff!L~&mn.n
AFG = = =
1
~titS:! t~1Jif{JlUHfi1M AFG 0
muntittj92 : tlMSlt~Ln~
=
5xl4+5xl4+5xl4
= 3(5xl4)= 3x70= 21
l3
AFE-
2FExAE-3.l5 2l X 2 X 6.3 - 3.15
= 3.!5cm "
= 10-8 = 2cm
2
&
= 3.1
'1
t~Ln~L~tmn.n
AGE
: ABCD
dlTIHru9J1.fll"dtf.Hl5 u
= 9cm, AD= 6cm S~ BK = 8cm '1
'i:ll8 AB ~Hlll
£fl :
ts~tffHl.flt~~dltutrulilJLffl"d
~msg ~~Uiltflts~~:mrult~RruHh~ru
'1
u
a
~
':l>
C1
n
•
s=
t s g t?inffits?l ru1 m t~tn~trts~ &m ru1 t ~ t~ rut'iY»~ R1J~m hltlm s tr:a '1
'i5
u
1
ms =
'i5 ir1 m U'l
"
xd ' B
~~Lf'iltflu~tfflrult
fflrul
ABCD =
t~Uiltfltfltfflrul
ABD+
sA seD
=
l
t~tnltJl
BDC '1
1
1Ao x BD + 1co x BD
D
= !BD(AO + OC) 2
1 !:!UtS g S u
=
xAC =
1 xd'
xd' B
G%1J~1Yt~~ l3cm s~
?irul£fl~~
14cm '1
u "" tSuRtfflrul&f\.HS"Jg
'>
1
'1
""
D 'i:ll ::::n~~G%'Hi'~~irl n
0
Li
&
u
l3cm s~ 14cm
firul6'Hi:!lli5 S 90
u
•
=
l
xd"
1 2
=-X
13
X
14 = 9lcm
2
cl
bh '1
~-,
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