Com.Mathematics (VIII)

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क ा ८ 

 
 
 
 
 
 

 

नेपाल सरकार  

िश ा ,िव ान तथा िविध म ालय

पा म िवकास के    

सानोिठिम, भ पुर     

ul0ft

sIff *

k|sfzs
gk] fn ;/sf/
lzIff, lj1fg tyf k|ljlw dGqfno

kf7o\ jm| d ljsf; sG] b|

;fgf]l7dL, eStk/'

k|sfzs
g]kfn ;/sf/
lzIff, lj1fg tyf k|ljlw dGqfno

kf7\oj|md ljsf; sG] b|

;fgfl] 7dL, eStk/'

© ;jf{lwsf/ kf7o\ jm| d ljsf; s]Gb|

o; kf7o\ k':ts;DaGwL ;Dk0" f{ clwsf/ kf7o\ jm| d ljsf; sG] b| ;fgf]l7dL, eStk/' df lglxt /xs] f] 5 . kf7o\ jm| d
ljsf; s]Gb|sf] lnlvt :jLsl[ tlagf Jofkfl/s k|ofh] gsf nflu o;sf] k/' } jf cf+lzs efu x'ax' k|sfzg ug,{
kl/jtg{ u//] k|sfzg ug,{ s'g} ljBt' Lo ;fwg jf cGo k|ljlwaf6 /]s8{ ug{ / kl| tlnlk lgsfNg kfOg] 5}g .

ky| d ;+:s/0f M lj=;=+ @)&@

kl/dflht{ bf];|f] ;+:s/0f M lj=;+ @)&^

d'b0| fM hgs lzIff ;fduL| s]Gb| ln=
;fgf]l7dL, eStk'/ .

dN" o ?=

kf7\ok:' ts;DaGwL kf7sx¿sf sg' } klg k|sf/sf ;´¬ fjx¿ ePdf kf7o\ j|md ljsf;
s]Gb,| ;dGjo tyf k|sfzg zfvfdf k7fOlbg'xg' cg'/fw] 5 . kf7sx¿af6 cfpg]
;¬´fjx¿nfO{ s]Gb| xflb{s :jfut ub5{ . tkfOn“ ] lsg]sf] kf7\ok':tsdf sg' } ql' 6
ePdf glhssf] ljt/saf6 pSt k':ts ;f6g\ ;Sgx' 'g] 5 .

xfd|f] egfO

lzIffnfO{ p2]Zodn" s, Jofjxfl/s / ;d;fdlos agfpg kf7o\ jm| d ljsf; sG] b|n] ljBfno txsf
kf7o\ jm| dtyf kf7o\ k:' ts ljsf; tyf kl/dfh{g ug{] sfon{ fO{ lg/Gt/tf lbFb} cfPsf] 5 . ljBfyL{df
/fi6k« d|] , /fli6o« tf kl| tsf] ;dk0{ f / nfs] tflGqs k4ltnfO{ cfTd;ft ug]{ efjgfsf] ljsf; u/fO{ gl} tsjfg,\
cgz' fl;t, :jfjnDaL tyf l;hg{ zLn eO{ ;dfjz] L ;dfh lgdf0{ fdf ofu] bfg lbg ;Sg] Ifdtf ljsf; xg'
cfjZos 5 .pgLx¿df eflifs tyf ul0ftLo l;ksf ;fy} lj1fg, ;r" gf tyf ;~rf/ kl| jlw, jftfj/0f,
:jf:Yo tyf hg;ªV\ of;DaGwL cfwf/et" 1fg tyf hLjgfk] ofu] L l;ksf] ljsf; xg' h?/L 5 . To;} u/L
ljBfyL{x¿df snf tyf ;f}Gbo{ kl| tsf] cg'/fu / dfgjLo d"No dfGotf, cfbz{ tyf jl} zi6\ox¿k|ltsf]
;rt] tf ;lxt ltgsf] ;+/If0f, ;j+ wg{ ug]{ efjgfsf] ljsf; cfjZos 5 . ;dtf dn" s ;dfhsf]
lgdf{0fdf ;xofu] k'¥ofpg pgLx¿df ljleGg hfthflt, lnªu\ , efiff, wd,{ ;:+ s[lt / Ifq] nufotsf
ljljwtfx¿sf] ;Ddfg ug{] / dfgj clwsf/ tyf ;dflhs d"No dfGotf k|lt ;r]t eO{
lhDdj] f/L jxg ug]{ efjgfsf] ljsf; u/fpg' cfjZos 5 . plNnlvt cfjZostfnfO{ bl[ i6ut u/L
cfwf/e"t lzIff kf7\oj|md -sIff ^–*_, @)^( nfO{ d"n cfwf/dfgL lzIff;DaGwL ljleGg cfofu] sf
;'emfj, lzIfs, ljBfyL{ tyf cleefjsnufot lzIff;u“ ;Da4 ljleGg JolSt ;lDdlnt uf]i7L /
cGt/lj|mofsf lgisif{ / ljleGg ljBfnodf k/LIf0f u/L k|fKt k[i7kf]if0f ;d]tnfO{ ;d]6L of]
kf7\ok:' ts tof/ kfl/Psf] xf] .

kf7\ok':tsnfO{ o; :j¿kdf Nofpg] sfo{df s]Gb|sf sfo{sf/L lgb]{zs >L lbjfs/ 9'ª\u]n,
kf| =8f= dLgaxfb/' >]i7, 8f= afns[i0f /l~ht, kf| = 8f= nv] gfy zdf{, ;'/G] b| cfrfo{, js} '07 vgfn,
j?0f j}B, ljho aflgof, ufd] f >i] 7, 808kfl0f zdf,{ xd] /fh kf]v/n] , hLj/fh cfrfo{, /d]z cj:yL,
/fhG] b| b]jsf6] f, dg} f clwsf/L, /fhs'df/ dfyd] f, ;/:jtL cfrfon{ ufotsf dxfg'efjsf] ljzi] f of]
ubfg /x]sf] 5 . o;sf] efiff ;Dkfbg xl/k|;fb lg/fn} f tyf 6fOk ;l] 6ª / nc] fp6 l8hfOg ho/fd
sO' s“ n] af6 ePsf] xf] . kf7o\ k:' tsnfO{ cWofjlws tyf kl/dfhg{ u/L ks| flzt ug{ sfod{ f o; sG] bs| f
dxflgbz{] s 8f= nv] gfy kf8} n] , >L u0fz] k|;fb e66\ /fO{ / >L lrgfsd' f/L lg/f}nfsf] ofu] bfg
/xs] f] 5 . o; kf7o\ k':tssf] ljsf; tyf kl/dfhg{ sfod{ f ;+nUg ;ak} l| t kf7\oj|md ljsf; s]Gb|
wGojfb k|s6 ub5{ .

kf7o\ k:' tsnfO{ lzIf0f l;sfOsf] dxŒjk0" f{ ;fwgsf ¿kdf lnOG5 . o; kf7o\ k:' tssf] k|ofu] af6
kf7o\ jm| dåf/f nlIft ;Ifdtf xfl;n ug{ ljBfyL{nfO{ ;xofu] k'Ug] ckI] ff ul/Psf] 5 . kf7\ok':tsnfO{
;s;] Dd ljm| ofsnfkd'vL / ?lrs/ agfpg] k|oTg ul/Psf] 5 . o; kf7\ok:' tsnfO{ cem} kl/ist[
kfg{sf nflu lzIfs, ljBfyL,{ cleefjs, a'l4hLjL Pjd\ ;Dk"0f{ kf7sx¿sf] ;d]t dxŒjk0" f{ e"ldsf

/xg] xb'“ f ;Da4 ;a}sf] /rgfTds ;'emfjsf nflu kf7\ojm| d ljsf; sG] b| xflb{s cg/' f]w ub{5 .

gk] fn ;/sf/
lzIff, lj1fg tyf kl| jlw dGqfno
lj= ;+= @)&^
kf7o\ j|md ljsf; s]Gb|

ljifo;"rL ki[ 7;ª\Vof

PsfO zLifs{ 1
10
1. /v] f / sf]0f 27
2. lqe'h, rte' {'h / axe' 'hx¿ 38
3. lqe'hsf] cg'¿ktf / ;d¿ktf 45
4. j[Q 50
5. 7f]; cfs[ltx¿ 57
6. lgbz{] fª\sx¿ 68
7. Ifq] kmn / cfotg 77
8. :yfgfGt/0f 83
9. lbzfl:ylt / :s]n 8«Oª 93
10. ;dxÒ 100
11. k0Ò f{ ;ª\Vofx¿ 104
12. kÒ0ff{ª\sx¿ 110
13. cfg'kflts ;ªV\ ofx¿ 119
14. jf:tljs ;ª\Vofx¿ 129
15. cg'kft, ;dfg'kft / kl| tzt 136
16. gfkmf / gf]S;fg 141
17. Pl] ss lgod 148
18. ;fwf/0f Aofh 161
19. tYofªs\ zf:q 188
20. aLhLo cleJo~hsx¿ 193
21. 3ftfªs\ 210
22. ;dLs/0f, c;dfgtf / n]vflrq
pQ/dfnf

kf7

1 /v] f / sf0] f (Line and Angle)

1.0 kg' /jnfs] g (Review)

ljm| ofsnfk 1

Pp6f ;fbf sfuhsf] kfgf np] m . To;nfO{ Ps k6s 7f8fl] t/ / csf{] k6s 41
t];f]{lt/af6 58\s] u/L k6\ofpm . To;kl5 k6\ofPsf] efu 32
vfn] . Toxf“ lrqdf bv] fP h:t} bO' c{ f6] f /v] fx¿ cfk;df kl| tR5l] bt bV] g]
5f} . ca tL /v] fnfO{ l;;fsndn] tfg . Toxf“ sf0] fx¿ ags] f] bV] g] 5f} .
ltgLx¿nfO{ jm| dzM 1, 2, 3 / 4 gfds/0f u/ .

-s_ 1 / 2, 2 / 3, 3 / 4, 4 / 1 s:tf ks| f/sf hf8] L sf0] fx¿ xg' \ <

-v_ 1 / 3, 2 / 4 s:tf ks| f/sf hf8] L sf0] fx¿ xg' \ < ;fyLx¿;u“ 5nkmn u/ .

dfly -s_ sf hf8] f sf0] fx¿df Pp6} zLifl{ aGb' / Pp6f ;femf eh' f 5 . o:tf hf8] f sf0] fx¿ cf;Gg
sf0] fx¿ xg' \ .

To:t} -v_ df ePsf hf8] f sf0] fx¿dWo] Pp6f sf0] f csf{] sf0] fsf] ljk/Lt lbzflt/ 5 . oL sf0] fx¿
zLiffl{ edv' sf0] fx¿ xg' \ .

bO' { ;/n /v] fv08 cfk;df sf6b\ f aGg] hf8] L sf0] fx¿df pbu\ d laGb' Pp6} / Pp6f eh' f ;femf
5 eg] To:tf sf0] fx¿nfO{ cf;Gg sf0] fx¿ (adjacent angles) elgG5 . To:t} olb Pp6f sf0] f csf{]
sf0] fsf] ljk/Lt lbzfdf 5 eg] To:tf hf8] L sf0] fx¿nfO{ zLiffl{ edv' sf0] fx¿ (vertically opposite
angles) elgG5 .

1.1. bO' { ;/n /v] fx¿ cfk;df sf6b\ f aGg] hf8] L sf0] fx¿sf] ko| fu] fTds k/LIf0f

(Experimental Verification of Pair of Angles Formed by Intersecting Two Lines)

k/LIf0f 1 : cf;Gg sf0] fx¿sf] ofu] kmn (Sum of adjacent angles)

lbOPsf lrqx¿df ;/n /v] f AB / CD laGb' O df kl| tR5b] g ePsf 5g\ .

AD A AD
D O
O
CB CO CB
B
lrq -s_ lrq -u_
xfdf| ] ul0ft, sIff * lrq -v_

1

ca, kf| 6] S] 6/ ko| fu] u/L sf0] fx¿ gfk / tnsf] tflnsfdf e/ .

lrq g=+ ∠AOC ∠AOD ∠AOC + ∠AOD kl/0ffd

-s_

-v_

-u_

dflysf] ko| fu] af6 s] lgisif{ kfof} <

olb bO' c{ f6] f ;/n /v] fx¿ cfk;df kl| tR5b] g ePsf 5g\ eg] Ps hf8] f cf;Gg sf0] fx¿sf]
ofu] kmn 180° jf bO' { ;dsf0] f xG' 5 .

k/LIf0f 2 : zLiffl{ edv' sf0] fx¿sf] ;DaGw (Relation betweens opposite angles)

tn lrqdf ;/n /v] f PQ / ST laGb' R df kl| tR5b] g ePsf 5g\ .

ca, kf| 6] S] 6/sf] ko| fu] u/L lbOPsf sf0] fx¿sf] gfk np] m / tnsf] tflnsfdf e/ M

P PT P T
SR R
SR T
Q Q
SQ

lrq -s_ lrq -v_ lrq -u_

lrq g=+ ∠PRS ∠QRT ∠SRQ ∠PRT kl/0ffd

-s_

-v_

-u_

dflysf] tflnsfaf6 s] lgisif{ kfof,} cfcfkm\ gf] sfkLdf nv] .

bO' { cf6] f ;/n /v] fx¿ cfk;df kl| tR5l] bt xb“' f aGg] zLiffl{ edv' sf0] fx¿ (vertically opposite
angles) a/fa/ xG' 5g\ .

pbfx/0f 1 A
O
tn lbOPsf] lrqdf lgDglnlvt hf8] L sf0] fx¿ s:tf sf0] fx¿ xg' ,\ nv] M

-s_ ∠XOB / ∠BOY -v_ ∠AOY / ∠XOB X Y

-u_ ∠BOX / ∠AOX -3_ ∠AOX / ∠BOY B

2 xfdf| ] ul0ft, sIff *

;dfwfg
oxf“ ;/n /v] fx¿ AB / XY laGb' O df sfl6Psf 5g\ .
-s_ ∠XOB / ∠BOY cf;Gg sf0] fx¿ xg' \ . -v_ ∠AOY / ∠XOB zLiffl{ edv' sf0] fx¿ xg' \ .
-u_ ∠BOX / ∠AOX cf;Gg sf0] fx¿ xg' \ . -3_ ∠AOX / ∠BOY zLiffl{ edv' sf0] fx¿ xg' \ .

pbfx/0f 2

lbOPsf] lrqdf x / y sf] dfg kQf nufpm . P S
;dfwfg R O x°
oxf“ ∠ROQ = 50°
50° y°
∠ROQ = ∠POS -zLiffl{ edv' sf0] fx¿_
Q

∴ x = 50°

km] l/ x + y = 180° -cf;Gg sf0] fx¿sf] ofu] kmn = 180° xG' 5 ._

cyjf, 50° + y = 180°

cyjf, y = 180° - 50° = 130°

∴ y = 130°

cEof; 1.1

1. tn lbOPsf sf0] fx¿dWo] sg' sg' sf0] fx¿ cf;Gg sf0] fx¿ xg' \ / sg' sg' zLiffl{ edv' sf0] fx¿

xg' ,\ nv] M PS

-s_ ∠PTS / ∠STQ -v_ ∠PTR / ∠STQ T

-u_ ∠PTS / ∠RTQ -3_ ∠PTS / ∠PTR RQ

-ª_ ∠RTQ / ∠QTS Y
O
2. ;u“ s} f] lrqdf /v] fx¿ XX' / YY' laGb' O df kl| tR5b] g ePsf 5g\ .

ca lrqaf6 4 hf8] f cf;Gg sf0] f / bO' { hf8] f zLiffl{ edv' X’ X

sf0] fx¿sf] ;r" L agfpm .

3. tn lbOPsf lrqx¿df x sf] dfg kQf nufpm M Y’

-s_ A -v_ -u_ 3x°
x°
C 105° x° D 80°
E x°

B

xfdf| ] ul0ft, sIff * 3

4. tn lbOPsf lrqx¿df x / y sf] dfg kQf nufpm M

-s_ -v_ x° -u_ x°
70°
x° 45° 100°
y° 60° y°
y°

5. lbOPsf lrqx¿df x, y / z sf0] fx¿sf] dfg kQf nufpm M

-s_ -v_ -u_

x° y° y° y°
45° z° x° 45°

50° 50° x° z°
z°

1.2. bO' c{ f6] f ;/n /v] fx¿nfO{ Pp6f 5b] sn] sf6b\ f aGg] sf0] fx¿

(Angles formed by a Transversal with two Straight Lines)

;u“ s} f] lrqdf bO' { ;/n /v] fnfO{ Pp6f 5b] sn] sf6s] f] 5 . tnsf kZ| gx¿af/] ;fyLx¿;u“
5nkmn u/ M

– slt cf6] f sf0] fx¿ ags] f 5g\ <

– sg' sg' sf0] fx¿ aflx/L sf0] f xg' \ < 12

– sg' sg' sf0] fx¿ leqL sf0] fx¿ xg' \ < 34

oxf“ ∠1, ∠2, ∠7 / ∠8 aflx/L sf0] fx¿ xg' \ 56
eg] ∠3, ∠4, ∠5 / ∠6 leqL sf0] fx¿ xg' \ . 78

-s_ PsfGt/ sf0] fx¿ (Alternate Angles)

dflysf] lrqdf ∠3 / ∠6 nfO{ x/] .

∠3 / ∠6 5b] ssf] bj' l} t/ k/s] f 5g\ / bj' } leqL cgf;Gg sf0] fx¿ xg' \ . t;y{ oL sf0] fx¿nfO{
PsfGt/ sf0] fx¿ elgG5 .

o:t} csf{] hf8] L sf0] fx¿ sg' sg' xfn] fg\ <

bO' { ;/n /v] fnfO{ 5b] sn] sf6b\ f 5b] ssf] bj' l} t/ k/s] f leqL cgf;Gg sf0] fx¿nfO{ PsfGt/
sf0] fx¿ (alternate angles) elgG5 .

dfly lrqdf lbOPsf ∠3 / ∠6; ∠4 / ∠5 PsfGt/ sf0] fx¿ xg' \ .

4 xfdf| ] ul0ft, sIff *

-v_ ;ªu\ t sf0] fx¿ (Corresponding Angles)

dflysf] lrqdf ∠1 / ∠5 nfO{ x/] f“} .

bj' } sf0] fx¿ 5b] ssf] Psl} t/ k/s] f 5g\ . ∠1 aflx/L sf0] f xf] eg] ∠5 leqL sf0] f xf] . t;y{ ∠1
/ ∠5 nfO{ ;ªu\ t sf0] fx¿ elgG5 . dfly lrqdf slt hf8] L ;ªu\ t sf0] f xfn] fg\ < nv] .

bO' { ;/n /v] fnfO{ 5b] sn] sf6b\ f 5b] ssf] Psl} t/ k/s] f Pp6f leqL / csf{] aflx/L cgf;Gg
sf0] fx¿nfO{ ;ªu\ t sf0] fx¿ (corresponding angles) elgG5 .

dfly lrqdf lbOPsf ∠1 / ∠5; ∠2 / ∠6; ∠3 / ∠7; ∠4 / ∠8 ;ªu\ t sf0] fx¿ xg' \ .

-u_ jm| dfut leqL sf0] fx¿ (Co-interior Angles)

dflysf] lrq x/] f“} . ∠3 / ∠5 s:tf hf8] L sf0] fx¿ xg' \ <

bj' } leqL cgf;Gg sf0] fx¿ xg' \ / bj' } 5b] ssf] Psl} t/ k/s] f 5g\ . logLx¿nfO{ jm| dfut
leqL sf0] fx¿ elgG5 .

bO' { ;/n /v] fnfO{ Pp6f 5b] sn] sf6b\ f aGg] 5b] ssf] Psl} t/ k/s] f leqL cgf;Gg sf0] fx¿nfO{
jm| dfut leqL sf0] fx¿ (co-interior angles) elgG5 .

dflysf] lrqdf ∠3 / ∠5; ∠4 / ∠6 jm| dfut leqL sf0] fx¿ xg' \ .

1.2.1 bO' c{ f6] f ;dfgfGt/ /v] fnfO{ 5b] sn] sf6b\ f aGg] sf0] fx¿sf] ;DaGw

olb lrqdf AB / CD ;dfgfGt/ /v] fx¿ eP A EG B
dfly k:| tt' ul/Psf hf8] L sf0] fx¿sf] ;DaGw
s:tf] xfn] f tnsf k/LIf0fx¿af6 x/] f“} . CD
H
k/LIf0f 1 : PsfGt/ sf0] fx¿sf] ;DaGw F

lrqdf bO' c{ f6] f ;dfgfGt/ /v] fx¿ AB / CD nfO{ 5b] s EF n] jm| dzM laGb' G / H df sf6s] f] 5 .

E E B E
AG
AG B AG B
C D
C D D H
H
CH

FF F

lrq -s_ lrq -v_ lrq -u_

ca tn tflnsfdf lbOPsf sf0] fx¿ kf| 6] S] 6/sf] ko| fu] u//] gfk / tflnsfdf e/ M

lrq g=+ ∠ AGH ∠GHD kl/0ffd ∠BGH ∠GHC kl/0ffd
-s_
-v_ 5
-u_

xfdf| ] ul0ft, sIff *

– dflysf] tflnsfaf6 s] lgisif{ kfof} <
– lbOPsf hf8] L sf0] fx¿ s:tf ks| f/sf sf0] fx¿ xg' ,\ ;fyLx¿;u“ 5nkmn u/L lgisif{ kQf nufpm .

bO' c{ f6] f ;dfgfGt/ /v] fnfO{ Pp6f 5b] sn] sf6b\ f ags] f PsfGt/ sf0] fx¿ a/fa/ xG' 5g\ .

k/LIf0f 2 : jm| dfut leqL sf0] fx¿sf] ofu] kmn

lbOPsf] lrqdf bO' c{ f6] f ;dfgfGt/ /v] fx¿ PQ / RS nfO{ 5b] s LM n] jm| dzM laGb' X / Y df
sf6s] f] 5 .

LX Q P X L L Q
PS R Q PX

RY Y S RS
M M
Y
lrq -s_ M

lrq -v_ lrq -u_

ca, kf| 6] S] 6/sf] ;xfotfn] tn lbOPsf sf0] fx¿sf] gfk / tflnsfdf e/ M

lrq g=+ ∠PXY ∠XYR ∠PXY + ∠XYR ∠QXY ∠XYS ∠QXY + ∠XYS kl/0ffd

-s_

-v_

-u_

– dflysf] tflnsfdf ePsf hf8] f sf0] fx¿ s:tf ks| f/sf sf0] fx¿ xg' \ <
– hf8] L sf0] fx¿sf] ofu] kmn slt eof] <
– of] k/LIf0fsf] lgisif{ s] xfn] f, ;u“ s} f] ;fyL;u“ 5nkmn u/ .

bO' { ;dfgfGt/ /v] fx¿nfO{ 5b] sn] sf6b\ f aGg] jm| dfut leqL sf0] fx¿sf] ofu] kmn 180° jf bO' {
;dsf0] f xG' 5 .

k/LIf0f 3 : ;ªu\ t sf0] fx¿sf] ;DaGw

tnsf lrqx¿df bO' { ;dfdfGt/ /v] fx¿ EF / GH nfO{ 5b] s MN n] jm| dzM laGb' K / L df sf6s] f] 5 .

M F M M
EK EK
F EK F

GH G H G H
L L L
N
N N
lrq -s_
lrq -v_ lrq -u_

6 xfdf| ] ul0ft, sIff *

kf| 6] S] 6/sf] ko| fu] u/L tn tflnsfdf lbOPsf sf0] fx¿ gfk / tflnsfdf e/ M

lrq g=+ ∠MKE ∠KLG ∠MKF ∠KLH ∠EKL ∠GLN ∠FKL ∠HLN kl/0ffd

-s_ 55° 55°

-v_ 90° 90°

-u_ 140° 140°

dflysf] tflnsfdf s:tf ks| f/sf hf8] L sf0] fx¿ 5g\ <

dflysf] k/LIf0faf6 s] lgisif{ lgsfNg ;lsG5, lgisif{ nv] L ;fyLx¿lar 5nkmn u/ .

bO' { ;dfgfGt/ /v] fx¿nfO{ 5b] sn] sf6b\ f aGg] ;ªu\ t sf0] fx¿ a/fa/ xG' 5g\ .

dflysf k/LIf0fx¿af6 xfdLn] lgDglnlvt s/' fx¿ klg yfxf kfpg ;S5f“} M

bO' { ;/n /v] fnfO{ Pp6f 5b] sn] sf6b\ f aGg,]
– PsfGt/ sf0] fx¿ a/fa/ ePdf
– jm| dfut leqL sf0] fx¿sf] ofu] kmn 180° ePdf jf
– ;ªu\ t sf0] fx¿ a/fa/ ePdf
tL bO' { ;/n /v] fx¿ ;dfgfGt/ xG' 5g\ .

pbfx/0f 1

;u“ s} f] lrqdf bO' c{ f6] f ;dfgfGt/ /v] fx¿nfO{ 5b] sn] sf6s] f] 5 . o;df a, b x, y, z sf] dfg kQf nufpm .

;dfwfg x° 125°
oxf“ x + 125° = 180° -cf;Gg sf0] fx¿_ z°
cyjf, x = 180° – 125°
cyjf, x = 55° y° b°
km] /L, x = y -;ªu\ t sf0] fx¿_ a°
cyjf, y = x = 55°
lrqcg;' f/, y = z = 55° -PsfGt/ sf0] fx¿_

∴ z = 55°

z + b = 180 -jm| dfut leqL sf0] fx¿_

55° + b = 180°

∴ b = 180° - 55° = 125° 7

cGTodf z = a -;ªu\ t sf0] fx¿_

∴ a = 55°

xfdf| ] ul0ft, sIff *

cEof; 1.2

1. ;u“ s} f] lrqdf bO' c{ f6] f ;/n /v] fnfO{ Pp6f 5b] sn] sf6s] f] 5 . pSt lrqaf6 lgDglnlvt

hf8] f sf0] fx¿sf] ;r" L tof/ kf/ M 8
76
-s_ aflx/L sf0] fx¿ -v_ leqL sf0] fx¿
5
-u_ PsfGt/ sf0] fx¿ -3_ jm| dfut leqL sf0] fx¿
34
-ª_ ;ªu\ t sf0] fx¿ 2

1

2. tn lbOPsf lrqx¿ x, y / z sf] dfg kQf nufpm -bO' c{ f6] f ;dfgfGt/ /v] fnfO{ 5b] sn] sf6s] f] 5_ M

-s_ -v_
z° x°
y° 60° y°
z°

x° 80°

-u_ -3_

2x 3x y° x°

z° 130°

-ª_ -r_

x° y° x° y°
47° z° 105°

-5_ -h_

3x°+10 y°
4x°-10 z°

8 60° x°

xfdf| ] ul0ft, sIff *

3. sf0] fx¿sf] gfksf cfwf/df tnsf bO' { /v] fx¿ AB / CD cfk;df ;dfgfGt/ 5g\ jf 5g} g,\
sf/0f;lxt nv] M

-s_ -v_ B D -u_

A 100° B 75° 75° B

A 64°

75° D C C 65° D
C A

4. tn lbOPsf lrqx¿df a, b, c, x, y, z sf] dfg kQf nufpm M

-s_ -v_

49° a°
y°
b°

x° x° y°
100°
-u_ x°
40° -3_
y°
z°
z° a° y°
50° 38° x°

-ª_ -r_

50° 50° 40°
y°
y° z°
a° x° z° x°

45° -h_

-5_

a° y° x° z° 75° 115° 58°
x° z°

y° a°

xfdf| ] ul0ft, sIff * 9

2kf7 lqeh' , rte' h{' / axe' h' x¿
(Triangle, Quadrilateral and Polygons)

2.0. kg' /jnfs] g (Review)

sDtLdf slt cf6] f l;wf l;Gsfx¿ ko| fu] u//] aGb cfsl[ t agfpg ;lsPnf <
bO' c{ f6] faf6 ;Dej 5 jf tLgcf6] f g} rflxG5, tLgcf6] f l;wf /v] fv08x¿n]
ags] f] aGb cfsl[ tnfO{ lqeh' elgG5 . tLgcf6] f eh' fx¿dWo] ;a} a/fa/ 5g\
eg] pSt lqeh' nfO{ ;dafx' lqeh' (equilateral triangle) elgG5 . olb sg' }
bO' c{ f6] f eh' fx¿ a/fa/ eP pSt lqeh' nfO{ s:tf] lqeh' elgG5 < eh' fx¿
km/s km/s gfksf eP lqeh' nfO{ s:tf] lqeh' elgG5 < lqeh' nfO{ eh' fsf
cfwf/df juLs{ /0f u/] h:t} sf0] fsf cfwf/df slt ks| f/df ljefhg ug{ ;lsPnf <

ca xfdL lqeh' sf ljleGg u0' fx¿sf] k/LIf0fsf af/d] f cWoog ub5{ f“} .

2.1 lqeh' sf u0' fx¿sf] k/LIf0f (Verification of Properties of Triangles)

k/LIf0f 1 : lqeh' sf leqL sf0] fx¿sf] ofu] kmn A P B
21 3
lrqdf bO' c{ f6] f ;dfgfGt/ /v] fx¿nfO{ 5b] s PQ /
5b] s PR n] sf6L ΔPQR ags] f] 5 . lqeh' sf tLg CQ 23 RD
sf0] fnfO{ jm| dzM 1, 2 / 3 dfgf“} . ca, AB//CD ePsfn]

∠APQ = ∠PQR xG' 5 / ∠BPR = ∠PRQ xG' 5 . -PsfGt/ sf0] fx¿ ePsfn_]

t;y,{ ∠APQ = ∠2 / ∠BPR = ∠3 xG' 5 .

ca, laGb' P df ∠1, ∠2 / ∠3 ldn/] l;wf sf0] f ∠APB agfp5“ .

l;wfsf0] f ∠APB sf] dfg slt xG' 5 < o;sf] dfg 180° xG' 5 . ∠1 + ∠2 + ∠3 = 180°

csf{] tl/sf M Pp6f afSnf] sfuhdf lqeh' lvr / sf0] fx¿nfO{ jm| dzM 1, 2 / 3 gfd bp] m . pSt
lqeh' nfO{ sr“} Ln] sf6 . To;kl5 zLifs{ f0] f A nfO{ BC df kg{] u/L k6o\ fpm . km] l/ zLifl{ aGb' B / C nfO{
klg A df gvK6g] u/L k6o\ fpm .

A

1

B2 3C B2 1 21 3
10 3C
xfdf| ] ul0ft, sIff *
A

ca, lqeh' ABC sf ltg zLifl{ aGbn' ] Pp6f l;wf sf0] f agfP . lrqdf bv] fP h:t} pSt l;wfsf0] fsf]
dfg 180° xG' 5 .

ko| fu] fTds k/LIf0f

km/s km/s gfksf eh' fx¿ ePsf tLgcf6] f lqeh' x¿ lvr . A

AA

BC B C BC

lrq -s_ lrq -v_ lrq -u_

kf| 6] S] 6/sf] ko| fu] u/L dflysf kT| os] lqeh' df ;a} sf0] fx¿ gfk / tnsf] tflnsf e/ M

lrq g=+ ∠BAC ∠ABC ∠ACB ∠BAC + ∠ABC + ∠ACB kl/0ffd
-s_

-v_
-u_

dflysf] tflnsfaf6 s] lgisif{ kfof} <
lqeh' sf leqL tLgcf6] f sf0] fsf] gfksf] ofu] kmn slt kfof} <
;u“ s} f] ;fyL;u“ lgisifa{ f/] 5nkmn u/ .

lqeh' sf leqL sf0] fx¿sf] gfksf] ofu] kmn 180° jf bO' { ;dsf0] f xG' 5 .

k/LIf0f 2 : ;dlåafx' lqeh' sf cfwf/sf sf0] fx¿sf] ;DaGw

km/s km/s cfwf/ ePsf tLgcf]6f ;dlåafx' lqe'hx¿ PQR lvr . h;df cfwf/ QR /
PQ = PR 5 .

kf| 6] S] 6/ ko| fu] u//] kT| os] lqeh' sf sf0] fx¿ gfk / tnsf] tflnsf e/ M

P
PP

Q RQ R R

lrq -s_ lrq -v_ Q

lrq -u_

xfdf| ] ul0ft, sIff * 11

lrq g=+ ∠PQR ∠PRQ ∠QPR kl/0ffd

-s_

-v_

-u_

dflysf] tflnsfaf6 s] lgisif{ kfof,} ;fyLx¿;u“ 5nkmn u/ .

;dlåafx' lqeh' sf cfwf/sf sf0] fx¿ a/fa/ xG' 5g\ .

k/LIf0f 3 : ;dlåafx' lqeh' sf zLifl{ aGba' f6 cfwf/sf] dWo laGbd' f lvlrPsf] /v] f / cfwf/sf] ;DaGw

1. km/s km/s cfwf/ ePsf tLgcf6] f ;dlåafx' lqeh' x¿ ΔXYZ lvr .

2. cfwf/ YZ sf] dWolaGb' P kQf nufO{ zLifl{ aGb' X ;u“ hf8] .

3. kf| 6] S] 6/sf] ko| fu] u/L ∠XPY / ∠XPZ gfk / tnsf] tflnsf e/ M

XX X

Y ZY P Z YZ
P P
lrq -v_
lrq -s_ lrq -u_

lrq g=+ ∠XPY ∠XPZ kl/0ffd

-s_

-v_

-u_

dflysf] tflnsfaf6 s] yfxf kfof} <

cf;Gg sf0] fx¿ a/fa/ eP jf dfg 90° eP s] xG' 5, ;fyLx¿;u“ 5nkmn u/ .

;dlåafx' lqeh' df zLifl{ aGba' f6 cfwf/sf] dWo laGb' hf8] g\ ] /v] f cfwf/;u“ nDa xG' 5 .

k/LIf0f 4 : ;dafx' lqeh' sf sf0] fx¿sf] ;DaGw

;jk{ y| d km/s gfksf tLgcf6] f ;dafx' lqeh' x¿ lvr .

12 xfdf| ] ul0ft, sIff *

kf| 6] S] 6/sf] ;xfotfn] ;a} sf0] fx¿ gfk / tnsf] tflnsf e/ M M

LM

LL

M N N N

lrq -s_ lrq -v_ lrq -u_

lrq g=+ ∠MLN ∠MNL ∠LMN kl/0ffd

-s_

-v_

-u_

dflysf] tflnsfaf6 s] lgisif{ lgsfNg ;lsG5, cfkm\ gf] sfkLdf nv] / ;fyLx¿;u“ 5nkmn u/ .

;dafx' lqeh' sf ;a} leqL sf0] fx¿ a/fa/ xG' 5g\ / kT| os] sf] dfg 60° xG' 5 .

k/LIf0f 5 : ;dsf0] fL ;dlåafx' lqeh' sf cfwf/sf sf0] fx¿sf] ;DaGw

Pp6f sf0] f ;dsf0] f (90°) ePsf km/s gfksf tLgcf6] f ;dlåafx' lqeh' ABC lvr .

lrqdf ABC ;dsf0] fL ;dlåafx' lqeh' xf] h;df AB = BC 5 / ∠B = 90° 5 .

A B BC

B C A C

lrq -s_ lrq -v_ A

lrq -u_

ca, kf| 6] S] 6/ ko| fu] u/L ;a} lqeh' sf cfwf/ sf0] fx¿ gfk / tnsf] tflnsfdf e/ M

lrq g=+ ∠BAC ∠ACB kl/0ffd

-s_

-v_

-u_

dflysf] tflnsfsf cfwf/df lgisif{ cfkm\ gf] pQ/ kl' :tsfdf nv] .

;dsf0] fL ;dlåafx' lqeh' sf cfwf/sf sf0] fx¿ 45° sf xG' 5g\ .

xfdf| ] ul0ft, sIff * 13

pbfx/0f 1 P

tnsf lqeh' x¿df afs“ L sf0] fx¿ kQf nufpm M

AL

40° x+20

B 55° C M N Q 2x x° R

-s_ -v_ -u_
;dfwfg

-s_ oxf“, ΔABC ;dlåafx' lqeh' xf] . o;df AB = AC / ∠ABC = 55° 5 . ΔABC ;dlåafx'
ePsfn] ∠ABC = ∠ACB = 55° xG' 5 .

∴ ∠ACB = 55°

km] l/, ∠ABC + ∠BCA + ∠CAB = 180° -lqeh' sf leqL sf0] fx¿sf] ofu] kmn = 180° xG' 5 ._
cyjf, 55° + 55° + ∠CAB = 180°
cyjf, ∠CAB = 180° - 110° = 70°

∴ ∠CAB = 70°

-v_ ΔLMN ;dsf0] fL lqeh' xf] . o;df ∠L = 40°; ∠M = 90° 5 .

∠L + ∠M + ∠N = 180°

cyjf, 40° + 90° + ∠N = 180°
cyjf, ∠N = 180°–130°

∴ ∠N = 50°

-u_ ΔPQR ljifdeh' -laifdafx_' lqeh' xf] .
∠P + ∠Q + ∠R = 180° -lqeh' sf leqL sf0] fx¿sf] ofu] kmn = 180° xG' 5 ._
cyjf, x + 20° + 2x + x = 180°
cyjf, 4x + 20° = 180°
cyjf, 4x = 180° - 20° = 160°
cyjf, 4x = 160°

∴ x = 40°

ca, ∠P = x + 20° = 40° + 20° = 60°

∠Q = 2x = 2 x 40° = 80°

∠R = x = 40°

14 xfdf| ] ul0ft, sIff *

cEof; 2.1

1. tLgcf6] f lqeh' lvrL ko| fu] åf/f kd| fl0ft u/ .

-s_ lqeh' sf] sg' } bO' { eh' fsf] nDafOsf] hf8] t;] f| ] eh' feGbf nfdf] xG' 5 .

-v_ lqeh' sf] Pp6f eh' f nDAofpb“ f aGg] aflx/L sf0] f leqL cgf;Gg sf0] fx¿sf] ofu] kmn;u“
a/fa/ xG' 5 .

2. tnsf lrqx¿df x, y, z sf] dfg kQf nufO{ afs“ L sf0] fx¿sf] dfg kQf nufpm M

-s_ P -v_ A -u_ X

x° x° 44°

Q 60° 30° R B y° C Y y° x° Z

-3_ E -ª_ Q -r_

20° 20° x° y°
x°
x° y° G P y° z° R 150°
F H
-h_ -em_
-5_
60°
x° 120° x° x°
2x° 6x° z°
45° y°
y°

-`_ -6_ x°

z° 38° 65°
x°
y°
y° 42° 60°

-7_ -8_

y° z° y°
45° x° 2x+5°

2x° 2x-5°
x°

xfdf| ] ul0ft, sIff * 15

3. ;u“ s} f] lrqdf AB||CD, ∠FEB = 85° / A C
∠AEC = 50° eP ∠ECF, ∠DFE / ∠CFE F
sf] dfg kQf nufpm . 50°

E
85°

BD

4. km/s gfkdf tLgcf6] f lqeh' ABC lvr h;df AC > AB 5 . ca tnsf] h:t} tflnsf
agfO{ lbOPsf eh' f / sf0] fsf] gfk e/ / lgisif{ nv] M

lrq AC AB ∠ACB ∠ABC kl/0ffd

-s_

-v_

-u_

lgoldt axe' h' sf] /rgf (Construction of Regular Polygons)

tnsf kZ| gx¿af/] 5nkmn u/f“} M

axe' h' egs] f] s] xf,] lgoldt axe' h' eGgfn] s] al' emG5, lgoldt axe' h' sf] leqL sf0] fx¿ s;/L kQf
nufpg ;lsG5 <

lgoldt axe' h' , axe' h' sf] leqL / aflx/L sf0] fx¿sf af/d] f xfdLn] sIff & df cWoog u/L
;ss] f 5f“} . oxf“ xfdL lgoldt axe' h' sf] /rgfsf af/d] f cWoog ub5{ f“{} .

olb lgoldt axe' h' sf] eh' fsf] ;ªV\ of n ePdf pSt axe' h' sf] leqL sf0] fsf] gfk n  2 1800 xG' 5 .
n

(I) lgoldt k~reh' sf] /rgf (construction of regular pentagon)

5cm eh' f nDafO ePsf] Pp6f lgoldt k~reh' sf] /rgf u/ . D
108°
klxnf] tl/sf
E 108° C
1. lgoldt k~reh' sf] leqL sf0] fsf] dfg kQf 108°
nufpg] tl/sf

oxf,“ n = 5 A 108° 108°
leqL sf0] f = B

16 5cm

xfdf| ] ul0ft, sIff *

2. AB = 5cm ePsf] l;wf /v] fv08 lvr / kf| 6] S] 6/n] laGb' B df 108° sf] sf0] f lvr .

3. pSt /v] fdf 5cm sf] rfkn] sf6 / C gfd bp] m . C df 108° sf] sf0] f lvr . ca 5cm df
lrxg\ nufO{ D gfd bp] m .

4. o:t} u/L laGb' D df 108° sf] sf0] f lvr / 5cm df lrxg\ nufO{ E gfd bp] m . clg laGb' E
/ A hf8] .

ca cfjZos k~reh' ABCDE tof/ eof] .

bf;] f| ] tl/sf

5cm Jof; ePsf] jQ[ leq k~reh' sf] /rgf u/ .

1. sDkf;sf] ;xfotfn] ;jk{ y| d Jof; AB = 5cm C
/ sG] b| O ePsf] Pp6f jQ[ lvr .

2. AB sf] nDafws{ lvr / jQ[ sf] kl/lw;Dd G F OE K
nDAofO jm| dzM C / D gfd bp] m . A B

3. km] l/ cwJ{ of; OB sf] nDafws{ lvr
/ sfl6Psf] laGbn' fO{ E gfd bp] m .

4. laGb' E af6 EC a/fa/sf] rfkn] OA
df sf6 / F gfd bp] m .

5. F af6 C a/fa/sf] rfk np] m / jQ[ sf] kl/lwdf H J
D
C af6 rfkx¿ lvr / jm| dzM G, H, J / K gfd bp] m .

6. ?n/n] C, G, H, J / K laGbx' ¿ hf8] .

7. ca cfjZos lgoldt k~reh' CGHJK tof/ eof] .

(II) lgoldt if6e\ h' sf] /rgf (construction of regular hexagon) -gfk/] x/] _

Pp6f eh' fsf] gfk 4cm ePsf] lgoldt if6e\ h' sf] /rgf u/ . C D

tl/sf M

1. AF = 4cm sf] Pp6f ;/n /v] f lvr .

2. laGb' A af6 / laGb' F af6 AF a/fa/ gfksf] B

rfk lnP/ sf6 / rfk lvr / sfl6Psf] OE

laGbn' fO{ O gfd bp] m .

3. O nfO{ cfwf/ dfg/] OA cwJ{ of; ePsf] jQ[ lvr .

4. OA a/fa/sf] rfkn] jQ[ sf] kl/lwdf A af6 B xb“' } jm| dzM F
sf6 / sfl6Psf] laGbn' fO{ jm| dzM B, C, D, E gfd bp] m . A
17
xfdf| ] ul0ft, sIff *

5. ca A, B, C, D, E / F nfO{ ?n/n] hf8] .

6. cfjZos if6e\ h' ABCDEF tof/ eof] .

(III) lgoldt ci6eh' sf] /rgf (construction of regular octagon)

5cm Jof; ePsf] jQ[ leq lgoldt ci6eh' sf] /rgf u/ .

tl/sf

1. sDkf;sf] ko| fu] u//] sG] b| O / Jof; AB = 5cm C
ePsf] Pp6f jQ[ lvr . E

2. Jof; AB sf] nDafws{ lvr . G

3. ca, ∠COA / ∠BOC sf] cws{ lvr . A OB

To;nfO{ kl/lw;Dd nDAofpm . ca sfl6Psf

laGbx' ¿nfO{ jm| dzM E / F tyf G / H gfd bp] m .

4. ?n/ ko| fu] u/L laGbx' ¿ jm| dzM A, E, C, G, B, F, F

D, H / A hf8] . H

5. ca cfjZos lgoldt ci6eh' tof/ eof] . D

gf6] M lgoldt if6e\ h' / lgoldt ci7eh' sf] /rgf klg k~reh' sf] h:t} leqL sf0] fx¿ kQf nufP/
klg ug{ ;lsG5 .

cEof; 2.2

1. leqL sf0] f kQf nufO{ tnsf gfksf] lgoldt k~reh' sf] /rgf u/ M

-s_ Pp6f eh' fsf] nDafO 4cm ePsf] -v_ Pp6f eh' fsf] nDafO 6cm ePsf]

2. leqL sf0] f kQf nufO{ tnsf gfksf] lgoldt if6e\ h' sf] /rgf u/ M

-s_ AB = 5cm -v_ eh' f = 6cm

3. sDkf;sf] ko| fu] u/L Pp6f eh' f 7cm ePsf] lgoldt if6e\ h' sf] /rgf u/ .

4. leqL sf0] f kQf nufO{ kf| 6] S] 6/sf] ko| fu] af6 lgDgfg;' f/ eh' f ePsf] lgoldt ci6eh' sf]
/rgf u/ M

-s_ 4cm -v_ 5cm -u_ 6cm

5. sDkf; / ?n/sf] ko| fu] u/L tn lbOPcg;' f/sf lgoldt axe' h' x¿sf] /rgf u/ M

-s_ cwJ{ of; 4cm ePsf] jQ[ leq lgoldt k~reh'

-v_ eh' sf] nDafO 5.5cm ePsf] if6e\ h'

-u_ Jof; 5cm ePsf] jQ[ leq lgoldt ci6eh'

18 xfdf| ] ul0ft, sIff *

2.3 ;dfgfGt/ rte' h{' , ju{ / cfotsf u0' fx¿sf] k/LIf0f

(i) ;dfgfGt/ rte' h{' sf u0' fx¿sf] vfh] L P Q
R
tnsf kZ| gx¿sf cfwf/df 5nkmn u/f+} M
B
;dfgfGt/ rte' h{' egs] f] s] xf] < S
o;sf u0' fx¿ s] s] xg' \ <

o;sf af/d] f xfdLn] cl3Nnf sIffx¿df cWoog ul/;ss] f 5f“} .
oxf,“ xfdL ;dfgfGt/ rte' h{' df lgDglnlvt u0' fx¿sf] k/LIf0f ub5{ f“} M

-s_ ;dfgfGt/ rte' h{' sf ;Ddv' sf0] fx¿ a/fa/ xG' 5g\ .

-v_ ;dfgfGt/ rte' h{' sf ;Ddv' eh' fx¿sf] gfk a/fa/ xG' 5 .

-u_ ;dfgfGt/ rte' h{' sf jm| dfut sf0] fx¿ kl/k/" s xG' 5g\ .

-3_ ;dfgfGt/ rte' h{' sf ljs0fx{ ¿ cfk;df ;dlåefhg xG' 5g\ .

-s_ ;dfgfGt/ rte' h{' sf ;Ddv' sf0] fx¿ a/fa/ xG' 5g\ .

km/s km/s gfk / lsl;dsf ltg cf6] f ;dfgfGt/ rte' h{' x¿ lvr .

A BA BA

D C C DC
D
lrq -s_ lrq -u_
lrq -v_

ca kT| os] ;dfgfGt/ rte' h{' sf sf0] fx¿ gfk / tnsf] h:t} tflnsf agfO{ k:| tt' u/ M

lrq g+ ∠DAB ∠BCD ∠ABC ∠CDA kl/0ffd
-s_

-v_

-u_

ca cfkmg\ f] lgisifn{ fO{ cfkm\ gf] ;dx" df 5nkmn u/L sIffdf k:| tt' u/ .

;dfgfGt/ rte' h{' sf ;Ddv' sf0] fx¿ a/fa/ xG' 5g\ .

-v_ ;dfgfGt/ rte' h{' df ;Ddv' eh' fx¿sf] gfk a/fa/ xG' 5 .

dfly g=+ -s_ df h:t} km/s km/s gfk / lsl;dsf tLg tLgcf6] f ;dfgfGt/ rte' h{' PQRS

lvr . cfkm" n] lvrs] f ;dfgfGt/ rte' h{' sf eh' fx¿ ?n/sf] ;xofu] df gfk / tnsf] h:t}

tflnsf agfO{ e/ M

xfdf| ] ul0ft, sIff * 19

P QP QP Q

S RS R S R

lrq -s_ lrq -v_ lrq -u_

lrq g=+ PQ RS QR PS kl/0ffd

-s_
-v_
-u_

ca, cfkmg\ f] lgisifa{ f/] cfkmg\ f] ;dx" df 5nkmn u/ / lgisif{ lgsfn .

;dfgfGt/ rte' h{' sf ;Ddv' eh' fx¿sf] gfk a/fa/ xG' 5 .

-u_ ;dfgfGt/ rte' h{' sf ljs0fx{ ¿ k/:k/ ;dlåefhg xG' 5g\ .

AB A B B
O O A O

DC D C C
D
lrq -s_ lrq -v_ lrq -u_

tLg tLg hgfsf] ;dx" lgdf0{ f u/L kT| os] n] km/s km/s gfksf tLg tLg cf6] f ;dfgfGt/
rte' h{' x¿ ABCD lvrL ljs0fx{ ¿ AC / BD hf8] / ljs0fx{ ¿sf] kl| tR5b] g laGbn' fO{ O dfg .

ca, kT| os] n] ;dfgfGt/ rte' h{' sf] ljs0fx{ ¿sf] gfk lng] / tnsf] h:t} tflnsf agfO{ k:| tt'
u/ M

lrq g=+ OA OC kl/0ffd OB OD kl/0ffd

-s_
-v_
-u_

ca, cfkmg\ f] lgisifn{ fO{ ;dx" df k:| tt' ug{] / ;dx" df 5nkmn u/L lgisif{ kQf nufpm .

;dfgfGt/ rte' h{' sf ljs0fx{ ¿ k/:k/ ;dlåefhg xG' 5g\ .

-ii_ cfotsf u0' fx¿sf] k/LIf0f

cfot egs] f] s] xf] / o;sf u0' fx¿ s] s] xg' ,\ o;sf af/d] f xfdLn] cl3Nnf sIffx¿df
cWoog ul/;ss] f 5f“} . ca xfdL o;sf lgDglnlvt u0' fx¿sf] k/LIf0f ug{] 5f“} M

20 xfdf| ] ul0ft, sIff *

cfot egs] f] Pp6f sf0] f 90° ePsf] ;dfgfGt/ rte' h{' xf] . t;y,{ ;dfgfGt/ rte' h{' sf ;a}

u0' fx¿ cfotsf klg u0' fx¿ xg' \ . o;sf ;fy,}

-s_ cfotsf ljk/Lt eh' fx¿ a/fa/ / ;dfgfGt/ xG' 5g\ . A B
-v_ cfotsf ;a} sf0] fx¿ a/fa/ / ;dsf0] fL xG' 5g\ .

-u_ cfotsf ljs0fx{ ¿ ;dlåeflht xG' 5g\ . D C
-3_ cfotsf ljs0fx{ ¿ a/fa/ xG' 5g\ .

dflysf u0' fx¿dWo] -s_ / -u_ xfdLn] ;dfgfGt/ rte' h{' df g} k/LIf0f ul/;ss] f 5f,“} o;nfO{
;dfgfGt/ rte' h'{ sf] ;66\ fdf cfot /fvL k/LIf0f u//] lzIfsnfO{ bv] fpm .

-v_ cfotsf ;a} sf0] fx¿ a/fa/ / ;dsf0] fL xG' 5g\ .

km/s km/s gfksf tLgcf]6f cfot lvr . cfk"mn] lvr]sf] cfotsf ;a} sf]0fx¿nfO{
kf| 6] S] 6/n] gfk/] tnsf] h:t} tflnsfdf k:| tt' u/ M

A BA BA B

D C DC D C

lrq -s_ lrq -v_ lrq -u_

lrq ∠DAB ∠ABC ∠BCD ∠CDA lgisif{
-s_
-v_
-u_

cfkmg\ f] lgisifa{ f/] ;fyLx¿;u“ 5nkmn u/ .

cfotsf ;a} sf0] fx¿ a/fa/ / ;dsf0] fL xG' 5g\ .

-3_ cfotsf ljs0fx{ ¿ a/fa/ xG' 5g\ .

km/s km/s gfksf tLgcf6] f cfotx¿ lvr / ljs0fx{ ¿ PR / QS hf8] .

P QP QP Q

S RS R SR

lrq -s_ lrq -v_ lrq -u_

xfdf| ] ul0ft, sIff * 21

ca ?n/ ko| fu] u/L ;a} ljs0fx{ ¿ gfkL tnsf] h:t} tflnsfdf k:| tt' u/ / lgisif{ klg kQf
nufpm M

lrq g=+ PR QS kl/0ffd

-s_

-v_

-u_

cfkmg\ f] lgisifa{ f/] ;fyLx¿lardf 5nkmn u/ .

cfotsf ljs0fx{ ¿ a/fa/ xG' 5g\ .

(iii) jus{ f u0' fx¿sf] k/LIf0f

ju{ egs] f] s] xf,] o;sf u0' fx¿ s] s] xg' ,\ o;sf af/d] f xfdLn] cl3Nnf sIffx¿df g} cWoog

ul/;ss] f 5f“} . ;a} eh' fx¿ a/fa/ ePsf] cfotnfO{ ju{ elgG5 . To; sf/0f cfotdf ;a}

u0' fx¿ jus{ f klg u0' fx¿ xg' \ . AB

jus{ f u0' fx¿ lgDgfg;' f/ 5g\ M

-s_ jus{ f ;Ddv' eh' fx¿ a/fa/ xG' 5g\ . DC
-v_ jus{ f ljs0fx{ ¿ a/fa/ xG' 5g\ .

-u_ jus{ f ;a} sf0] fx¿ a/fa/ / ;dsf0] fL xG' 5g\ .

-3_ jus{ f ljs0f{ x¿ cfk;df ;dsf0] fL xg' ] u/L ;dlåeflht xG' 5g\ .

-ª_ jus{ f kT| os] ljs0fn{ ] zLif{ sf0] fnfO{ cfwf ub5{ g\ .

dflysf -s_, -v_ / -u_ sf u0' fx¿ cfot / ;dfgfGt/ rte' h{' sf u0' fx¿;u“ ldNbfhN' bf 5g\ .
t;y{ oL u0' fx¿ cufl8 u/] h:t} k/LIf0f u/L lzIfsnfO{ bv] fpm .

-3_ jus{ f ljs0fx{ ¿ cfk;df ;dsf0] f xg' ] u/L ;dlåeflht xG' 5g\ .

km/s km/s gfksf tLgcf6] f jux{ ¿ lvrL lrqdf bv] fP em“} ljs0fx{ ¿ lvr . clg ljs0fx{ ¿

sfl6Psf] 7fpn“ fO{ 0 gfd bp] m . K L
L
K LK

O OO

N M NM N

lrq -s_ lrq -v_ M

22 lrq -u_
xfdf| ] ul0ft, sIff *

ca ?n/ / kf| 6] S] 6/ ko| fu] u//] tn lbOPsf sf0] fx¿ / eh' fx¿ gfk / tnsf] tflnsfdf h:t} agfO{
k:| tt' u/ M

lrq sf0] fx¿sf] gfk ljs0f{ KM sf ljs0f{ LN sf kl/0ffd

v08x¿ v08x¿

∠KOL ∠LOM ∠MON ∠NOK OK OM OL ON

-s_
-v_
-u_

ca dflysf] tflnsfaf6 s] lgisif{ lgsfNg ;lsG5 nv] / ;fyLx¿;u“ 5nkmn u/L ;dx" df lgisif{
lgsfn .

jus{ f ljs0fx{ ¿ cfk;df ;dsf0] f xg' ] u/L ;dlåefhg xG' 5g\ .

-ª_ jus{ f kT| os] ljs0fn{ ] zLifs{ f0] fx¿nfO{ cfwf u5g{ \ . A B

tLg tLg hgfsf] ;dx" agfpm / ;an} ] Ps Pscf6] f O
ju{ ABCD df ljs0fx{ ¿ AC / BD lvr .

DC

ca kT| os] n] tnsf] h:t} tflnsf agfO{ lbOPsf sf0] fx¿ gfk / tflnsfdf e/ M

zLifs{ f0] fsf] gfk ;xfos sf0] fsf] gfk lgisif{
∠ABO ======== / ∠CBO =
∠ABC = ∠BCO = ====== / ∠OCD =
∠BCD = ∠CDO = ====== / ∠ODA =
∠CDA = ∠DAO = ====== / ∠OAB =
∠DAB =

tflnsfsf cfwf/df lgsfns] f] cfkmg\ f] lgisifn{ fO{ ;dx" df k:| tt' u/ / 5nkmn u/L ;fdl" xs
lgisif{ lgsfn .

ju{ ABCD df ljs0f{ AC n] zLifs{ f0] f ∠DAB / ∠BCD nfO{ cfwf u/s] f] 5 . To:t,} ljs0f{ BD n]
zLifs{ f0] fx¿ ∠ABC / ∠CDA nfO{ cfwf kf/s] f] 5 .

jus{ f kT| os] ljs0fn{ ] zLifs{ f0] fx¿nfO{ cfwf ub5{ g\ .

xfdf| ] ul0ft, sIff * 23

pbfx/0f 1

lbOPsf lrqx¿df x, y / z sf] dfg kQf nufpm M

-s_ -v_ -u_ P 45° Q

A BK L
x°
10 cm z°
O
120° y O
D CN
MS R

;dfwfg
-s_ ABCD Pp6f ;dfgfGt/ rte' h{' xf] .

oxf,“ ∠BCD = 120° 5 .
∠BAD = x = <
xfdLnfO{ yfxf 5, ;dfgfGt/ rte' h{' sf ;Ddv' sf0] fx¿ a/fa/ xG' 5g\ .

∠BAD = ∠BCD

cyjf, x = 120°
-v_ KLMN Pp6f cfot xf] h;df

ljs0f{ KM = 10cm 5 eg] ljs0f{ LN = ycm 5 .
xfdLnfO{ yfxf 5, cfotsf ljs0fx{ ¿ a/fa/ xG' 5g\ . t;y{ KM = LN xG' 5 .

∴ y = KM = 10cm xG' 5 .
-u_ PQRS Pp6f ju{ xf] h;df PR / QS bO' c{ f6] f ljs0fx{ ¿ 5g\ .

∠OQP = 45° 5 / ∠OQR = Z 5 .
xfdLnfO{ yfxf 5, jus{ f ljs0fn{ ] zLifs{ f0] fnfO{ cfwf u5{ . t;y{

∠OQR = ∠OQP xG' 5 -lsgls QS ljs0f{ xf_]

∴ ∠OQR = Z = 45° x'G5 .

cEof; 2.3.

1. ;dfgfGt/ rte' h{' , ju{ / cfotsf u0' fx¿sf] ;r" L tof/ kf/ .
2. cfot / jus{ f km/s u0' fx¿ s] s] 5g,\ kQf nufpm .

24 xfdf| ] ul0ft, sIff *

3. tnsf lrqx¿df x,y / z sf] dfg kQf nufpm M

-s_ -v_ -u_ P Q

A BP 5 cm x Q y°
x° RS x°
CS
70°
D

R

-3_ -ª_ -r_

A 8cm B K 4cm 3cm L E x° y° F
z° G
6cm y yox
D
x CN MH

4. nDafO (l)= 18cm / rf}8fO (b) = 9cm ePsf] Pp6f cfot agfO{ To;sf u'0fx¿sf]
k/LIf0f u/ .

2.4. cfotsf] /rgf (Construction of Rectangle)

-s_ bO' { ljs0fx{ ¿ / ltgLx¿larsf] sf0] f lbOPdf,

ljs0fx{ ¿ AC = BD = 7cm / ∠COD = 45° ePsf] cfotsf] /rgf u/ .

r/0fx¿

1. AC = 7cm sf] Pp6f l;wf /v] f lvr . X
D
2. AC sf] dWo laGb' O kQf nufpm .
C
3. sDkf;sf] ko| fu] n] laGb' O df 45° sf]
sf0] f lvr / XY ;Dd nDAofpm . 25

4. OA a/fa/sf] rfkn] X lt/ / Y lt/ sf6 A

/ jm| dzM D / B gfd bp] m . O

5. ?n/sf] ko| fu] u/L laGbx' ¿ A, B, C /
D jm| dzM hf8] .

6. cfjZos cfot ABCD tof/ eof] . B
Y

xfdf| ] ul0ft, sIff *

-v_ Pp6f eh' f, Pp6f ljs0f{ / To;n] ToxL eh' f;u“ agfPsf] sf0] f lbOPdf

cfwf/ /v] f BC = 8 cm, ∠DBC = 30° / ljs0f{ BD = 9 cm ePsf] cfotsf] /rgf u/ .

r/0fx¿

1. BC = 8 cm sf] Pp6f cfwf/ /v] f lvr . Y

2. sDkf;sf] ko| fu] u/L B df 30° sf] sf0] f X
lvr / X ;Dd nDAofpm .
A D
3. sDkf;df 9 cm nfdf] rfk lnP/ BX df C
sf6 / D gfd bp] m .

4. C / D nfO{ ;/n /v] fn] hf8] .

5. B df sDkf;sf] ;xfotfn] 90° sf] sf0] f B 30°
lvr / BY /v] f tfg .

6. BY df CD a/fa/sf] rfkn] sf6 / A gfd bp] m .

7. laGb' A / D hf8] .

8. cfjZos cfot ABCD tof/ eof] .

cEof; 2.4

1. tnsf kT| os] cj:yfdf cfotsf] /rgf u/ M

-s_ ljs0f{ (AC) = BD = 8cm, ∠BOC = 30° -v_ ljs0f{ (PR) = 7cm, ∠QOR = 45°

-u_ ljs0f{ (BD) = 10cm, ∠AOD = 60°

2. bO' { ljs0fs{ f] larsf] sf0] f 75° ePsf] / ljs0fs{ f] nDafO 7.4cm ePsf] Pp6f cfotsf]

/rgf u/ .

3. tnsf kT| os] cj:yfdf cfotsf] /rgf u/ M

-s_ ljs0f{ PR = 6 cm PQ = 3 cm, ∠QPR = 60° ePsf] cfot PQRS.

-v_ BC = 7.1 cm, BD = 10 cm, ∠DBC = 45° ePsf] cfot ABCD.

-u_ Pp6f eh' f 4.8 cm / ljs0f{ 6.2 cm

-3_ AC = 5 cm, AB = 4 cm / ∠BAC = 60° ePsf] cfot ABCD.

4. tnsf kT| os] cj:yfdf cfotsf] /rgf u/ M

-s_ ljs0f{ AC = 8cm / AC / BD larsf] sf0] f 45° ePsf]

-v_ Pp6f ljs0fs{ f] nDafO 7cm / bO' l{ js0f{ larsf] sf0] fsf] gfk 30° ePsf]

-u_ Pp6f eh' f 5cm, ljs0f{ 10cm / tL bO' l{ arsf] sf0] f 60° ePsf]

-3_ PR = 9.9cm, PQ = 7cm / QPR = 45° ePsf] .

26 xfdf| ] ul0ft, sIff *

kf7 lqeh' sf] cg¿' ktf / ;d¿ktf

3 (Congruence and Similarity of Triangles)

3.0 kg' /jnfs] g ( Review)
tnsf tLg hf]8f cfs[ltx¿df s] s] ;dfgtf / s] s] km/s b]lvG5, ;fyLx¿;“u 5nkmn u/ M

klxnf] hf8] f lqeh' x¿df bj' } p:t} cfsf/ / Pp6} gfksf 5g\ . t;y{ oL bO' { lqeh' x¿ cg¿' k 5g\ .
bf;] f| ] hf8] f ;dfgfGt/ rte' h{' x¿df bj' } p:t} cfsf/ t/ km/s gfksf 5g\ . t;y{ oL bO' { rte' h{' x¿
;d¿k 5g\ . To:t,} t;] f| ] hf8] f lrqx¿ bj' } km/s km/s cfsf/ / km/s gfksf 5g\ . t;y{ tL bO' {

cfsl[ tx¿ cg¿' k klg 5g} g\ / ;d¿k klg 5g} g\ .

sg' } bO' { HofldtLo cfsl[ tx¿ p:t} cfsf/ / Pp6} gfksf 5g\ eg] A P

To:tf HofldtLo cfsl[ tx¿nfO{ cg¿' k (congruent) elgG5 . R

B CQ

lrqdf ΔABC / ΔPQR cg¿' k 5g\ . ;ªs\ t] df ΔABC ≅ ΔPQR nl] vG5 .

sg' } bO' { HofldtLo cfsl[ tx¿ p:t} cfsf/sf 5g\ eg] A P
R
To:tf HofldtLo cfsl[ tx¿nfO{ ;d¿k (similar) elgG5 . CQ

B

lrqdf ΔABC / ΔPQR ;d¿k 5g\ . ;ªs\ t] df ΔABC ∼ ΔPQR nl] vG5 .

3.1. lqeh' cg¿' k xg' ] cj:yfx¿sf] k/LIf0f

ljm| ofsnfk 1. s'g} ΔABC lbOPsf] 5 . pSt lqeh' ;u“ cg¿' k xg' ] u/L slt tl/sfn] /rgf ug{
;lsG5, x/] f“} M

tl/sf 1
oxf“ ΔABC lbOPsf] 5 . ;jk{ y| d BC a/fa/sf] gfk ePsf] cfwf/ A
A'
/v] f B'C' lvr . laGb' B af6 AB a/fa/sf] sDkf;sf] rfkn] C'

dfly rfk lvr . km] l/ C af6 CA a/fa/sf] rfkn] sf6 / B C B'

sfl6Psf] laGbn' fO{ A' n] hgfpm .

A', B' / A',C' hf8] . lrqcg;' f/ ΔABC sf tLgcf6] f eh' f / ΔA'B'C' sf tLgcf6] f eh' f;u“ jm| dzM
a/fa/ 5g\ . ΔABC / ΔA'B'C' cg¿' k 5g\ . -gfk/] x/] _

xfdf| ] ul0ft, sIff * 27

o;/L Pp6f lqeh' df tLgcf6] f eh' fx¿ csf{] lqeh' sf tLgcf6] f A X
CY
eh' fx¿;u“ cnu cnu cfk;df a/fa/ 5g\ eg] pSt bO' {

lqeh' x¿ cg¿' k xG' 5g\ . Z

o;nfO{ eh' f eh' f eh' f (side, side, side) jf B

5f6] s/Ldf e=' e=' e=' (SSS) tYo elgG5 . lrqdf ΔABC ≅ ΔXYZ 5 .

tl/sf 2 X
P'
QR sf] nDafO a/fa/sf] cfwf/ /v] f Q'R' lvr . P

sDkf;sf] ko| fu] u/L ∠Q gfk / ToxL a/fa/sf] sf0] f Q'

df lvr / /v] f Q'X tfg .

ca QP a/fa/sf] rfkn] Q' af6 Q'X df Q R Q' R'

sf6 / P' gfd bp] m .

laGb' P' / R' hf8] . o;/L ags] f] ΔP'Q'R' / ΔPQR cg¿' k xG' 5g\ . t;y{ ΔPQR ≅ ΔP'Q'R' xG' 5 .

olb Pp6f lqeh' df bO' c{ f6] f eh' fx¿ / ltgLx¿larsf] P P'
R'
sf0] f;u“ csf{] lqeh' sf bO' c{ f6] f eh' fx¿ / ltgLx¿larsf]

sf0] f cnu cnu cfk;df a/fa/ 5g\ eg] tL

bO' { lqeh' x¿ cg¿' k xG' 5g\ . QR Q'
o;nfO{ eh' f sf0] f eh' f (side, angle, side)

5f6] s/Ldf e=' sf=] e=' (SAS) tYo elgG5 . oxf“ ΔPQR ≅ ΔP'Q'R' 5 .

tl/sf 3 A A'
C B' C'
BC a/fa/sf] nDafO ePsf] cfwf/ B'C' lvr .
kf| 6] S] 6/sf] ko| fu] u/L ∠B gfk / B' df ToxL
gfksf] sf0] f lvr . km] l/ C sf] dfg
kf| 6] S] 6/sf] ko| fu] u/L gfk / ;fx] L a/fa/sf] sf0] f B
C' df lvr . bO' { /v] f sfl6g] laGbn' fO{ A' gfd bp] m .
ca ΔABC / ΔA'B'C' cg¿' k xG' 5g\ . -gfk/] x/] _

Pp6f lqeh' sf] bO' c{ f6] f sf0] f / tL sf0] fx¿larsf] eh' f, csf{] lqeh' sf] bO' c{ f6] f sf0] f / tL

sf0] fx¿larsf] eh' f;u“ cfk;df cnu cnu a/fa/ eP tL lqeh' x¿ cg¿' k xG' 5g\ .

o;nfO{ sf0] f, eh' f, sf0] f (Angle, side, angle) 5f6] s/Ldf sf=] e=' sf=] (ASA) tYo elgG5 .

ΔABC ~= ΔA'B'C' xG' 5 . A A'

B C B' C'

28 xfdf| ] ul0ft, sIff *

tl/sf 4

olb, ΔABC ;dsf0] fL lqeh' ePdf pSt lqeh' ;u“ cg¿' k xg' ] lqeh' s;/L /rgf ug{ ;lsG5, x/] f“} M

1. oxf“ ΔABC Pp6f ;dsf0] fL lqeh' 5 . o;df ∠B ;dsf0] f (90°) 5 .

2. BC a/fa/ xg' ] u/L B'C' cfwf/ A X
/v] f lvr . A'

3. sDkf;÷kf| 6] S] 6/sf] ko| fu] u/L B'
df 90° sf] sf0] f lvr .

4. ΔABC sf] s0f{ AC a/fa/sf] rfk B C B' C'
C' af6 lnP/ lrqdf B'X df sf6 /
sfl6Psf] laGbn' fO{ A' gfd bp] m .

5. A' / C' hf8] . ca ΔABC / ΔA'B'C' cg¿' k 5g,\ gfk/] x/] .

Pp6f lqeh' sf ;dsf0] f, s0f{ / Pp6f eh' fsf cfwf/df klg ΔABC ;u“ cg¿' k lqeh' /rgf ug{ ;lsof] .

Pp6f lqeh' sf] ;dsf0] f, s0f{ / Pp6f eh' f csf{] lqeh' sf] P X
;dsf0] f, s0f{ / Pp6f eh' f cfk;df cnu cnu a/fa/ 5g\ Q
eg] lqeh' cg¿' k xG' 5g\ . o;nfO{ ;dsf0] f, s0f{ / RY X
eh' f (right angle, hypotenues / side) 5f6] s/Ldf
;=s=e=' (R.H.S) tYo elgG5 . lrqdf ΔPQR ~= ΔXYZ

pbfx/0f 1

lbOPsf hf8] f lqeh' x¿ cg¿' k 5g\ . x sf] dfg lgsfnL afs“ L sf0] f / eh' fx¿sf] gfk kQf nufpm M

;dfwfg C

oxf“ ABC / XYZ cg¿' k 5g\ . A 4.3cm
∠A = ∠X = 35°, B = ∠Y = 123° / ∠C = ∠Z = 22° (1.7x-315.3°)cm123B°
km] l/ AB = XY X

cyjf, (1.7x - 1.3)cm = (3x-3.9)cm 4.9cm (3x-3.9)cm
cyjf, 1.7x - 1.3 = 3x - 3.9
cyjf, 3x - 1.7x = 3.9 - 1.3 Z 22° 123°

Y

cyjf, 1.3x = 2.6

cyjf, x = 2.6  2
1.3

xfdf| ] ul0ft, sIff * 29

To; sf/0f, AB = 1.7x - 1.3 = 1.7 x 2 - 1.3 = 2.1cm

XY = 3x - 3.9 = 3 x 2 - 3.9 = 2.1cm
AC = XZ = 4.9cm
BC = YZ = 4.3cm

cEof; 3.1

1. tnsf hf8] L lqeh' x¿sf eh' fx¿ tyf sf0] fx¿ gfk / cg¿' k 5g\ jf 5g} g,\ nv] M
-s_ -v_

2. tnsf hf8] L lqeh' x¿ sg' tYosf cfwf/df cg¿' k 5g,\ nv] M
-s_ -v_

-u_ -3_

3. tnsf cg¿' k lqeh' x¿df ;ªu\ t eh' f / ;ªu\ t sf0] fx¿ 56' o\ fP/ nv] M

-s_ P L N -v_ X A
60°
60° 50° 65°
B
50° R 60° Z C 60°
65°

Q 5cm M Y

4. lbOPsf cg¿' k lqeh' x¿df x / y sf] dfg kQf nufO{ yfxf gePsf eh' f / sf0] fx¿sf]
dfg lgsfn M

-s_ 70° 48° -v_ y+10°
x°
(5y-3)° 62°
122° 38°
70° (4x-4)°

30 xfdf| ] ul0ft, sIff *

-u_ A Q -3_ L

56°(3x-0.6)cm 3.1cm Y 3cm
(2x+1)cm 70°
64°
C 2.5cmN
60° 45° Z X

P 56° 70°
B 64° M

(x+3)cm R 2y

A D
B
5. lbOPsf] lrqdf laGb' D /v] f AB sf] dWolaGb' xf]
/ CD ⊥ AB 5 eg] sg' tYosf cfwf/df C
ΔACD / ΔBCD cg¿' k 5g,\ bv] fpm M

PQ

6. ;u“ s} f lrqdf ΔPQR / ΔPQS nfO{ cg¿' k bv] fpm,
RS hxf“ ∠RPQ = ∠PQS / QS = PR 5 .

AP

7. ΔABC / ΔPQR df ∠A = ∠P / ∠B = ∠Q 5 .

tnsf dWo] sg' cj:yf ykk] l5 ~ΔABC = ΔPQR xG' 5 <

-s_ ∠C = ∠R -v_ AB = PQ -u_ BC = QR B CQ R

8. lrqdf sg' cj:yf ykk] l5 ΔLMN / ΔXYZ cg¿' k xG' 5g\ < X Z
hxf“ ∠M = ∠X = 90° 5 / MN = XZ 5 . L Y
sg' tYosf cfwf/df cg¿' k xG' 5g\ <

MN

xfdf| ] ul0ft, sIff * 31

3.2 ;d¿ktf (Similarity)

ljm| ofsnfk 1 P'

1. ?n/ / l;;fsndsf] ko| fu] u/L ΔPQR lvr . P
R Q'
2. sfkLsf] csf{] 7fpd“ f Pp6f l;wf/v] f Q'X lvr /

Q' af6 QR sf] bfA] a/ rfk nfO{ Q'X df sf6 R' X

/ R' gfd bp] m . Q

3. Q' af6 PQ sf] bfA] a/ rfk lnO{ dflylt/ sf6 / To;u} /L R' af6 PR sf] bfA] a/ rfkn] sf6 .
sfl6Psf] laGbn' fO{ P' gfd bp] m .

4. P', Q' / P', R' hf8] .

ca bj' } lqeh' sf sf0] fx¿ / eh' fx¿ gfk / tnsf] h:t} tflnsf agfO{ e/ M

ΔPQR ΔP'Q'R'

sf0] fsf] gfk ∠P = ∠Q = ∠R = ∠P' = ∠Q' = ∠R' =
P'R' =
eh' fsf] gfk PQ = QR = PR = P'Q' = Q'R' =

dflysf] tflnsfaf6 lgDglnlvt cgk' ftx¿ kQf nufpm M

PQ  QR  PR 
P'Q' Q'R' P'R'

∠P = ∠P' = ... ∠Q = ∠Q' = / ∠R = ∠R' = ........

dflysf] ljm| ofsnfkaf6 s] yfxf kfof} <

;ªu\ tL eh' fsf] cgk' ft s:tf] 5, ;ªu\ tL sf0] fx¿lar s] ;DaGw 5 <

cfkm\ gf] lgisif{ nv] / sIffsf7] fdf k:| tt' u/ .

oxf“ ΔPQR / ΔP'Q'R' sf ;ªu\ tL eh' f ;dfgk' flts 5g\ / sf0] fx¿sf] gfk a/fa/ 5 .
To;sf/0f ΔPQR / ΔP'Q'R' ;d¿k 5g\ . o;nfO{ ΔPQR ~ ΔP'Q'R' nl] vG5 .

;d¿k lqeh' df ;ªu\ tL eh' fx¿ ;dfgk' flts -cgk' ft a/fa/_ / ;ªu\ t sf0] fx¿ a/fa/ xG' 5g\ .

A olb Δ ABC ~Δ PQR eP,

P AB  BC  AC
PQ QR PR

∠A = ∠P ∠B = ∠Q ∠C = ∠R x'G5 .

B CQ R

32 xfdf| ] ul0ft, sIff *

3.2.1. lqeh' x¿ ;d¿k xg' ] cj:yfx¿ (Conditions for triangles to be similar)
-s_ bO' { hf8] f ;ªu\ tL sf0] fx¿ a/fa/ ePdf,
ΔABC 5 h;df ∠B = 30°, ∠C = 60° / BC = 6cm 5 . csf{] A'B'C' /rgf u/ h;df ∠B = 30°,
∠C' = 60° / B'C' = 4cm 5 .

A A'

30° 60° B' 30° 60°
B C'

6cm C 4cm

ca ΔABC / ΔA'B'C' sf eh' fx¿ gfk / tflnsfdf e/ M

AB/A'B' BC/B'C' AC/A'C' lgisif{

AB = BC = AC =

A'B' = B'C' = A'C' =

dflysf] tflnsfsf ;a} eh' fx¿sf] cgk' ft a/fa/ bl] vof] . ;ªu\ tL eh' fx¿ ;dfgk' flts 5g\ .
To;sf/0f ΔABC ~ ΔA'B'C' xG' 5 .

olb lqeh' x¿df bO' { hf8] L ;ªu\ t sf0] fx¿ a/fa/ 5g\ eg] ;ªu\ t eh' fx¿ klg ;dfgk' flts xG' 5g\
/ lqeh' x¿ ;d¿k xG' 5g\ . o;nfO{ sf=] sf=] (AA) sf] tYo elgG5 .

-v_ ltg cf6] } eh' fx¿ ;dfgk' flts ePdf,

ΔABC 5 h;df AB = 2cm, BC = 3cm / AC = 4cm 5 . ΔA'B'C' lvr, h;df A'B' =3cm,
B'C' = 4.5cm / A'C' = 6cm 5 .

A'

A 4cm 6cm
2cm
3cm

B C B' C'
3cm 4.5cm

dflysf ΔABC / ΔA'B'C' sf sf0] fx¿ gfk / tnsf] tflnsfdf e/ M

sf0] fx¿ lgisif{ lgisif{ lgisif{

∠A = ∠B = ∠C =

∠A' = ∠B' = ∠C' =

xfdf| ] ul0ft, sIff * 33

tLgcf6] } eh' fx¿ ;dfgk' flts ePdf ;ªu\ tL sf0] fx¿ klg a/fa/ xG' 5g\ . t;y{ lbOPsf lrqdf
ΔABC ~ ΔA'B'C' eof] . o;nfO{ e=' e=' e=' ;d¿ktf (SSS similarity) elgG5 .
-u_ bO' c{ f6] f eh' fx¿sf] cgk' ft / ltgLx¿larsf] sf0] f a/fa/ ePdf,

ΔABC lbOPsf] 5 h;df AB = 5cm, BC = 6cm / ∠ABC = 45° 5 . csf{] ΔXYZ /rgf u/
h;df XY = 2.5cm, YZ = 3cm / ∠XYZ = 45° 5 .

A

X
5cm

45° 2.5cm
B 45°
C Y Z
6cm
3cm

dflysf lqeh' df lgDgfg;' f/sf sf0] f / eh' fx¿ gfk / tflnsfdf e/ M

kl/0ffd

∠A = ∠B = ∠C = AC = AC
XZ =

∠X = ∠Y = ∠Z = XZ =

dfyLsf] tflnsfaf6 ;a} ;ªu\ t sf0] fx¿ a/fa/ eP / afs“ L eh' fsf] cgk' ft klg Pp6} cfof] .
t;y{ ΔABC / ΔXYZ ;d¿k eP . o;nfO{ eh' f, sf0] f, eh' f (SAS) tYo elgG5 .

olb hf8] f lqeh' df bO' { eh' fx¿sf] cgk' ft / ltgLx¿larsf] sf0] f a/fa/ ePdf tL bO' { lqeh' x¿ ;d¿k xG' 5g\ .

pbfx/0f 1

lbOPsf lqeh' x¿ ;d¿k 5g\ eg] x, y / z sf] dfg kQf nufpm .
;dfwfg
oxf“ ABC ~ PQR 5 . t;y{ eh' fx¿ ;dfgk' flts xG' 5g\ .

AB  BC  AC xG' 5 . A
PQ QR PR P

lrqfg;' f/ M 12  18  x .........(i) 12cm x y 15cm
y 6 15

klxnf] / bf;] f| ] cg' kft lnb“ f 60° z° R
C Q 6cm
12  18 B
y6 18cm

cyjf 18 x y = 12 x 6

cyjf y  12 6  4cm xfdf| ] ul0ft, sIff *
18
34

km] l/ bf;] f| ] / t;] f| ] cgk' ft lnb“ f

18  x
6 15

cyjf, 6x = 15 x 8

x= 15 x 8 = 20cm
6
/ ∠B = ∠Q xG' 5 .

∠B = 60° = Q
∴ ∠Z = 60°

pbfx/0f 2

;u“ s} f] lrqdf ΔLMN / ΔLEF ;d¿k 5g\ eg] EF / EM sf] dfg kQf nufpm .

;dfwfg

oxf,“ ΔLMN / ΔLEF ;d¿k 5g\ . L
4cm 4cm
t;y,{ LM  MN  LN xG' 5 .
LE EF LF

cyjf, y4  5  64 xG' 5 . E F
4 x4 x 6cm

klxnf] / t;] f| ] cgk' ft lnb“ f, y N

y  4  10 M
44 5cm

cyjf, y + 4 = 10
cyjf, y = 10 - 4 = 6cm
km] l/ bf;] f| ] / t;] f| ] cgk' ft lnb“ f,

cyjf, 10x = 20 35
cyjf, x =

∴ x = 2cm

xfdf| ] ul0ft, sIff *

pbfx/0f 3
lrqdf EF || GH / ∠EFO = ∠OGH 5 eg] kd| fl0ft u/ ΔEFO ~ ΔOGH

kd| f0f M oxf“ EF || GH 5 t;y{ EH 5b] s xf] . E
(1) ∠FEO = ∠GHO xG' 5 . -PsfGt/ sf0] f ePsfn_]

(2) ∠EFO = ∠OGH O G
H
lbPsf] F
(3) ∠EOF = ∠GOH -zLiffl{ edv' sf0] fx¿_
(4) ΔEFO / ΔGOH sf ;ªu\ tL sf0] fx¿ a/fa/ eP,
t;y,{ ΔEFO ~ ΔGHO xG' 5 .

cEof; 3.2

1. tn lbOPsf hf8] f lqeh' x¿sf] sf0] f / eh' fx¿ gfk / ;d¿k 5g\ jf 5g} g,\ kQf nufpm M
-s_ -v_ -u_

2. lbOPsf hf8] f ;d¿k lqeh' x¿df x, y / z sf] dfg kQf nufpm M

-s_ 8cm x -v_ 10.5cm
y 30°
10cm 8 cm z° 12cm
y
x 7.5cm

6cm 3cm

-u_ -3_ A 3cm B

21in y 1.5cm y
x
6in E x
24in 4cm

8in C 30° 60° D
A 6cm

3. lbOPsf] lrqdf ΔBEC ~ ΔADC 5 CD = 20cm

AD = 8.8cm / EC = 5cm 5 eg] BE sf] dfg 8.8cm B

kQf nufpm . 5cm
20cm E
D C

36 xfdf| ] ul0ft, sIff *

4. lbOPsf] lrqdf olb BC || DE / ∠CED = 30° 5 A
eg] -s_ ΔADE ~ ΔABC bv] fpm . -v_ DE / ∠ACB
4.5cm 6cm

sf] gfk kQf nufpm . B C
7.2cm 4cm

D E

P

8cm 5. lrqdf PS ⊥QR 5 / ΔPQR ~ ΔPSR 5 eg] PS
6cm sf] gfk kQf nufpm . olb ∠PQR = 30° eP,

Q 30° ∠RPS slt xfn] f <
R
16 cm S X

6. lrqdf ∠Y = ∠Z, XY = 20cm, A B
AY = 15.5cm / XZ = 15cm eP, Z
(i) ΔXAZ ~ ΔXBY bv] fpm . Y
L
(ii) XB sf] gfk kQf nufpm .

7. ;u“ s} f] lrqdf ML || NP 5 Q P
∠LMN = ∠NPQ = 90° 5 eg,] M N
ΔLMN ~ ΔNPQ bv] fpm .

xfdf| ] ul0ft, sIff * 37

kf7

4 jQ[ (Circle)

4.0. kg' /jnfs] g (Review)

tnsf jQ[ x¿df cªl\ st efu / 5fof kfl/Psf] efusf] gfd nv] / ;dx" df 5nkmn u/ M

-s_ -v_ -u_ -3_

OA O O A 111111111122222222223333333333444444444455555555556666666666777777777788888888889999999999000000000011111111112222222222333333333344444444445555555555666666666677777777778888888888 B
PQ
-ª_ -5_
-r_

O O
X Y L1111111111222222222233333333334444444444555555555566666666667777777777888888888899999999990000000000111111111122222222223333333333 M
jQ[ sf ljleGg efusf af/d] f xfdLn] cl3Nnf] sIffdf g} cWoog ul/;ss] f 5f“} . ca xfdL jQ[ sf]

kl/lw / Ifq] kmnsf af/d] f cWoog ub5{ f“} .

4.1. jQ[ sf] kl/lw / Jof;sf] ;DaGwsf] vfh] L

jQ[ sf] kl/lw (Circumference of Circle)

ljm| ofsnfk 1 O 2 cm P

5/5 hgfsf] ;dx" lgdf0{ f u/L kT| os] n] jm| dzM afSnf] sfuhdf jm| dzM 2 cm, 2.5
cm, 3 cm, 3.6 cm, 4 cm cwJ{ of; ePsf jQ[ lvr / sr“} Lsf] ;xfotfn] To;nfO{
sf6 . To;kl5 lrqdf bv] fP em} “
Pp6f cwJ{ of; lvr .

lrqdf bv] fP em} “ laGb' P ?n/sf]
ky| d /v] fdf kg{] u/L /fv / pSt
j[QnfO{ u'8fpm . pSt j[QnfO{
ta;Dd u'8fp ls laGb' P n] km]l/
:sn] sf] csf{] /v] fnfO{ 5fc] f;] \ . To;kl5
lrqdf bv] fP em“} ;?' sf] laGb' / clGtd laGb' l6kf6] u/ . tL bO' { laGbl' arsf] nDafO g} jQ[ sf] kl/lw
xG' 5 .

jQ[ sf] kl/lw (c) / Jof; (d) gfk, [ hxf“ Jof; (d) = 2r xG' 5 ] . ca jm| dzM tflnsfdf k:| tt' u/ M

38 xfdf| ] ul0ft, sIff *

;d"x jQ[ sf] Jof; (d) jQ[ sf] kl/lw (C) c
d

-s_ 4 cm

-v_ 5 cm 15.70 15.70  3.14
5

-u_ 6 cm

-3_ 7 cm

-ª_ 8 cm

o;/L dflysf] tflnsfaf6 s] k:| 6 xG' 5 eg] jQ[ sf] Jof; hlt;s' } ePtf klg pSt jQ[ sf] kl/lw /
Jof;sf] cgk' ft ;w“} 3.14 sf] cf;kf;df xG' 5 . t;y{ kl/lw / Jof;sf] cgk' ftnfO{ 3.14 dflgG5 .
jf C /d = 3.14 -crn /flz_ xG' 5 . o;nfO{ lus| cIf/ π (Pie jf kfO_{ n] hgfOG5 . o;nfO{

22 klg nl] vG5 .
7

To;sf/0f, c  xG' 5 .
d

∴C= πd hxf“   22 xG' 5 .
7

xfdLnfO{ yfxf 5, jQ[ sf] cwJ{ of; Jof;sf] cfwf xG' 5 . t;y,{ d = 2r, ∴ C = 2πr xG' 5 .

ljm| ofsnfk 2

3/3 hgfsf] ;dx" agfpm . To;kl5 kT| os] ;dx" n] km/s d
cfsf/sf an] gfsf/ j:tx' ¿ np] m . kT| os] ;dx" n] pSt d
d

an] gfsf/ j:ts' f] cfwf/sf] Jof; gfk . To;kl5 lrqdf

bv] fP h:t} u/L pSt an] gfsf/ j:ts' f] cfwf/sf] glhs

jl/kl/ Pp6f wfuf] afw“ . Tof] wfufs] f] nDafO pSt cfwf/

jQ[ sf] kl/lw a/fa/ xG' 5 . kT| os] ;dx" n] To;kl5 cfkmg\ f]

;dx" n] lnPsf] kl/lw -wfuf_] / Jof;sf] nDafOsf] cgk' ft k:| tt' u/ . pSt cgk' ft kf| oM ;ad} f Pp6}

3.14 sf] jl/kl/ cyft{ \ 22 kfOG5 . To;nfO{ g} π -kfO_{ elgG5 .
7

t;y{ π C -kl/lw_ nl] vG5 .
= d -Jof;_

∴ C = πd = 2πr [ d = 2r]

xfdf| ] ul0ft, sIff * 39

pbfx/0f 1

olb Pp6f jQ[ sf] Jof; 9 cm 5 eg] pSt jQ[ sf] kl/lw slt xfn] f <   22 
 7

;dfwfg

oxf“ lbPsf] jQ[ sf] Jof; (d) = 9 cm

jQ[ sf] kl/lw (C) = ?

xfdLnfO{ yfxf 5 ls C = πd /   22
7

C  22  9  28.30 ctM jQ[ sf] kl/lw (C) = 28.30cm xG' 5 .
7

pbfx/0f 2

Pp6f an] gfsf/ 6o\ fªs\ Lsf] cfwf/sf] kl/lw 471cm 5 eg] pSt 6o\ fªs\ Lsf] cfwf/sf] cwJ{ of; slt
xfn] f < -π = 3.14 ko| fu] ug{] ._

;dfwfg

oxf“ an] gfsf/ 6o\ fªs\ Lsf] cfwf/sf] kl/lw (C) = 471 cm

,, ,, ,, cwJ{ of; (r) = ?

xfdLnfO{ yfxf 5 kl/lw (c) = 2πr

cyjf, 471 cm = 2 × 3.14 × r

cyjf, 6.28 r = 471 cm

cyjf, r  471 cm = 75 cm.
6.28

ctM pSt 6o\ fªs\ Lsf] cwJ{ of; (r) = 75 cm xG' 5 .

cEof; 4.1

1. π = 3.14 ko| fu] u/L lbOPsf kT| os] jQ[ sf kl/lw kQf nufpm M

-s_ cwJ{ of; = 3cm -v_ Jof; = 5cm -u_ cwJ{ of; = 4.5 cm

-3_ Jof; = 10 inch -ª_ cwJ{ of; = 12m -r_ Jof; = 18ft.

2. π = 3.14 ko| fu] u/L lbOPsf] kl/lwsf cfwf/df jQ[ sf] cwJ{ of; kQf nufpm M

-s_ C = 12.56 cm -v_ C = 18.84 inch -u_ C = 34.54 cm

-3_ C = 65.94 ft. -ª_ C = 113.04 cm -r_ C = 376.8 yd

40 xfdf| ] ul0ft, sIff *

3. Pp6f jQ[ fsf/ vn] db} fgsf] cwJ{ of; 84 ld6/ eP pSt db} fgsf] kl/lw slt xfn] f <   22 
 7

4. cwJ{ of; 100 ld6/ ePsf] jQ[ fsf/ wfjg dfud{ f wfjsn] Ps rSs/ nufpb“ f slt ld6/
b/' L kf/ u5{ xfn] f < [ π = 3.14]

5. cfwf/sf] kl/lw 157 ft. ePsf] jQ[ fsf/ ejgsf] Jof; slt xfn] f < [ π = 3.14]

6. Pp6f jQ[ fsf/ g;/{ Lsf] Jof; 56 m. 5 . To;nfO{ aflx/af6 jl/kl/ af/ nufpg slt ld6/
tf/ rflxPnf, olb 704 m. tf/ pknAw 5 eg] slt k6s jl/kl/ tf/ af/ nufpg ;lsPnf <

  22 
 7

7. Pp6f df]6/;fOsnsf] rSsf 150 rSs/ nufp“bf 396 ld= b'/L kf/ u5{ eg] pSt

df6] /;fOsnsf] rSsfsf] Jof; slt xfn] f <   22 
 7

8. Pp6f an] gfsf/ sf7sf] 7s] Lsf] jl/kl/ 3 kmGsfd] f tf/n] afW“ bf 132 inch nfdf] tf/ rflxG5

eg] pSt 7s] Lsf] Jof; slt xfn] f <   22 
 7

4.2. jQ[ sf] Ifq] kmn (Area of circle) 34 56

ljm| ofsnfk 3 2 7P

cwJ{ of; OP ePsf] jQ[ sf] Jof;nfO{ cfwf/ dfg/] jQ[ nfO{ a/fa/ 16 1 o 8
efudf ljefhg u/ / 1 bl] v 16 gDa/ lbg] lrqdf bv] fP h:t} Jof;af6 16 9
10
15

dflysf efux¿df km/s /ªn] /ªu\ fpm . To;kl5 sr“} Lsf] ;xfotfn] 16 14 13 12 11

cf6] f efunfO{ sf6 . ;a} sfl6;sk] l5 Pp6f kl5 csf{] ub{} 15 6j' m| fx¿nfO{ lrq g=+ !
lrqdf bv] fP em}“ ldnfpm . To;kl5 clGtd 6j' m| fnfO{ a/fa/ bO' { efudf

ljefhg u/ / lrq g=+ 2 sf] h:t} bj' } k6l\ 6 ldnfP/ /fv . o;/L Pp6f jQ[ nfO{ cfotfsf/ ¿kdf

ldnfpg ;lsG5 .

h;df, nDafO (l) = kl/lwsf] cfwf 123 45 678

= 1 2r  r 9 10 11 12 13 14 15
2

rf8} fO (b) = jQ[ sf] cwJ{ of; = r 5 . lrq g+= @

ca, xfdLnfO{ yfxf 5,

cfotsf] Ifq] kmn (A) = nDafO (l) x rf8} fO (b)

= πr × r 41

= πr2 ju{ PsfO
To;n} ] jQ[ sf] Ifq] kmn (A) = π r2 ju{ PsfO

xfdf| ] ul0ft, sIff *

xfdLnfO{ yfxf 5, jQ[ sf] cwJ{ of; Jof;sf] cfwf xG' 5 . rd
2

t;y{, A   d 2
2

A  d2 ju{ PsfO

4

pbfx/0f 3

olb Pp6f jQ[ sf] Jof; 12cm 5 eg] pSt jQ[ sf] Ifq] kmn slt xfn] f < (π = 3.14)

;dfwfg
oxf,“ jQ[ sf] Jof; (d) = 12 cm.

cwJ{ of; r  d  12  6cm
22

ca, xfdLnfO{ yfxf 5 . jQ[ sf] Ifq] kmn (A) = π r2 ju{ PsfO 67ccmm

= 3.14 × 6× 6 cm2 O

= 113.04 cm2

pbfx/0f 4

lbOPsf] lrqdf 5fof kfl/Psf] efusf] Ifq] kmn kQf nufpm . (π = 3.14)

;dfwfg

oxf,“ ABCD Pp6f ju{ xf] . h;df AB = BC = 14 cm 5 .

ca, ju{ ABCD sf] Ifq] kmn (A1) = l2 = (14)2 cm2 A B
C
= 14 × 14 cm2 D
14 cm
= 196 cm2

kml] / oxf“ lrqdf 4 cf6] f Ps rfy} fO jQ[ x¿ 5g\ .

h;df cwJ{ of; (r)  14  7cm 5 .
2

t;y{ Ps rfy} fO jQ[ sf] Ifq] kmn  1 r2 ju{ PsfO
4

 1 (3.14) 7 7cm2
4

 1 (153.86)cm2
4

42 xfdf| ] ul0ft, sIff *

To:t,} 4 cf6] f rfy} fO jQ[ x¿sf] hDdf Ifq] kmn (A2) = 4 1 (153.86) cm2
4

= 153.86 cm2

ca, 5fof kfl/Psf] efusf] Ifq] kmn (A) = A1 - A2

= (196 - 153.86) cm2

= 42.14 cm2

pbfx/0f 5

olb Pp6f jQ[ fsf/ kf8} L kfv] /Lsf] kl/lw 125.6 m 5 eg] pSt kfv] /Lsf] lkw“ sf] cwJ{ of; / Ifq] kmn
kQf nufpm . (π = 3.14)

;dfwfg x
oxf“ jQ[ fsf/ kfv] /Lsf] kl/lw (C) = 125. 6 m

cwJ{ of; (r) = ?

ca, kl/lw (C) = 125.6 m

cyjf, 2πr = 125.6 m [∴ c = 2πr]

cyjf, 2 × 3.14 × r = 125.6

cyjf, r  125.6 m  20m
2  3.14

t;y,{ jQ[ sf] cwJ{ of; (r) = 20 m

cyf{t,

kfv] /Lsf] cwJ{ of; (r) = 20 m

kml] / jQ[ fsf/ kfv] /Lsf] lkw“ sf] Ifq] kmn = ?

xfdLnfO{ yfxf 5, jQ[ sf] Ifq] kmn (A) = πr2 ju{ PsfO

= 3.14 × 20 × 20 m2

= 1256m2

xfdf| ] ul0ft, sIff * 43

cEof; 4.2

1. tnsf jQ[ x¿sf] Ifq] kmn kQf nufpm M[π = 3.14]

-s_ cwJ{ of; = 3 cm. -v_ Jof; = 5 cm. -u_ cwJ{ of; = 8 ft.

-3_ Jof; = 12 inch -ª_ Jof; = 18m. -r_ Jof; = 20km.

-5_ Jof; = 15 mm -h_ Jof; = 22 cm. -em_ cwJ{ of; = 16 yd

2. olb Pp6f jQ[ fsf/ sf7] fsf] Jof; 14 ld6/ 5 eg] pSt sf7] fsf] Ifq] kmn slt xfn] f <[π = 3.14]

3. lgDg lnlvt kl/lw ePdf jQ[ sf] Ifq] kmn kQf nufpm M [π = 3.14]

-s_ 34.54 cm -v_ 65.94m -u_ 1884 inch

-3_ 113.04m -ª_ 376.80 ft.

4. Pp6f an] gfsf/ srf/} fsf] cfwf/sf] Jof; 9cm eP pSt srf/} fsf] cfwf/sf] ;txsf] Ifq] kmn
slt xG' 5 < (π = 3.14)

5. tnsf lrqx¿df 5fof kfl/Psf] efusf] Ifq] kmn lgsfn M

-s_ -v_ -u_ -3_ 28 cm

12 ft.

8cm

6. Pp6f an] gfsf/ 6o\ fªs\ Lsf] lkw“ sf] Ifq] kmn 154 ju{ lkm6 5 eg] pSt 6o\ fªs\ Lsf] kl/lw

/ cwJ{ of; kQf nufpm .   22 
 7

7. Pp6f 153.86 m2 Ifq] kmn ePsf] jQ[ fsf/ vn] db} fgnfO{ 9nfg ul/of] eg] pSt db} fgsf] 9nfg
u/s] f] efusf] Jof; slt xfn] f / pSt db} fgsf] 9nfgsf] 3/] f slt ld6/ xfn] f <

8. zldn{ fn] 5cm cwJ{ of; ePsf] Pp6f jQ[ lvlrg\ . To;u} /L ks| fzn] klg 7cm cwJ{ of; ePsf]
csf{] jQ[ lvr] . ca s;n] lvrs] f] jQ[ sf] Ifq] kmn w/] } 5 / sltn] w/] } 5 <

9. cfkmg\ f] sfkLdf Pp6f 7.5 cm cwJ{ of; ePsf] jQ[ lvr/] /ª nufpm . To;kl5 /ªu\ fPsf]
efusf] Ifq] kmn lgsfn .

44 xfdf| ] ul0ft, sIff *

kf7

5 7f;] cfsl[ tx¿ (Solid Shapes)

5.0 kg' /jnfs] g (Review) -s_ -v_

;u“ s} f] tflnsfdf x/] / lbOPsf
kZ| gx¿sf af/d] f 5nkmn u/ M

!= tflnsf -s_ df s:tf ks| f/sf
cfsl[ tx¿ 5g\ <

@= tflnsf -v_ df s:tf ks| f/sf cfsl[ tx¿ 5g\ <

#= tflnsf -s_ / -v_ df ePsf cfsl[ tx¿sf] gfdsf] ;r" L agfpm .

$= tflnsf -s_ / -v_ sf cfsl[ tx¿lar s] km/s 5 <

dflysf] tflnsfdf -s_ df ;a} ;dtnLo cfsl[ t (Plane Shapes) 5g\ eg] tflnsf -v_ df 7f;]
cfsl[ tx¿ /xs] f 5g\ . h;cGtut{ 3g (Cube), if8d\ v' f (cuboid), an] gf (Cylinder), ufn] f (Sphere),
;fn] L (Cone) sf af/] cl3Nnf] sIffdf g} cWoog u¥of“} . ca xfdL lqeh' sf/ lkH| d / lk/fld8x¿sf
af/d] f cWoog u/f“} .

5.1 lqeh' fsf/ lkH| d / lk/fld8 (Triangular Prism and Pyramid)

-s_ lqeh' fsf/ lkH| d (Triangular Prism)

sf8{ af8] d{ f Pp6f 12 cm nDafO / 8 cm rf8} fO ePsf] cfot agfpm . lrqdf bv] fP h:t}
nDafOtkma{ f6 a/fa/ tLg efudf ljefhg u/ . To;kl5 lrqdf bv] fP h:t} larsf]
efusf] tn / dfly ;dafx' lqeh'
agfpm . sr“} Ln] pSt cfsl[ tnfO{
sf6 / /v] fx¿af6 k6o\ fpm .

s:tf] cfsl[ t aGof,] x/] L tnsf kZ| gsf] pQ/ nv] M 45
-s_ o;df sltcf6] f lqeh' fsf/ ;tx 5g\ <
-v_ o;df sltcf6] f cfotfsf/ ;tx 5g\ <
-u_ o;sf] gfd s] xfn] f <

xfdf| ] ul0ft, sIff *