गणित कक्षा ७

13.4 s f]0fx¿sf] k|ofu] fTds k/LIf0f (Experimental Verification of

Angles)
tn lbOPsf laleGg sf]0fx¿sf] ko| fu] fTds k/LIf0f ug{x' f;] \ M

k/LIf0f 1

bO' c{ f]6f l;wf/v] fx¿ Ps cfk;df sfl6Fbf aGg] zLiffl{ ed'v sf]0fx¿ a/fa/ xG' 5g\ .

oxfF, RQ SQ

OO

PS PR

lrq 1 lrq 2

bO' { l;wf /]vfx¿ PQ / RS nfO{ laGb' O df sfl6g] u/L lvRgx' f;] \ . ca
k|f]6o\ fS6/sf ;xfotfn] sf0] fx¿ jm| dzM ∠ROQ, ∠QOS, ∠ROP / ∠POS nfO{
gfKg'xf];\ / tnsf] tflnsfdf eg{'xf];\ .

lrq ∠ROP ∠QOS ∠ROQ ∠POS kl/0ffd

1
2

lgisifM{ b'O{cf6] f l;wf/v] fx¿ Ps cfk;df sfl6Fbf aGg] zLiff{ledv'
sf]0fx¿ a/fa/ xG5 .

k/LIf0f 2

Pp6f l;wf/]vfn] csf]{ l;wf/]vf;Fu Ps}lt/ agfPsf cf;Gg sf0] fx¿sf] ofu] kmn

180° xG' 5 . PP

AQ BA QB

lrq 1 lrq 2

dflysf] lrqdf AB l;wf /v] fv08df s'g} laGb' Q af6 QP /v] fv08 lvrL km/s

194 ul0ft, sIff &

km/s b'Oc{ f6] f lrqx¿ lvRgx' f;] \ . ca rfbF sf] ;xfotfn] sf0] fx¿ j|mdzM ∠PQA /
∠PQB nfO{ gfKgx' f;] \ / tnsf] tflnsfdf eg{x' f;] \ .

lrq g=+ ∠PQA ∠PQB kl/0ffd

1
2

lgisif{ M Pp6f l;wf/v] fn] csf]{ l;wf/v] f;Fu Ps}lt/ agfPsf cf;Gg
sf]0fx¿sf] of]ukmn 180° xG' 5 .

k/LIf0f 3
s'g} laGb'sf] jl/kl/ Ps kl/jm| d0fdf ags] f sf]0fx¿sf] of]ukmn 360° xG' 5 .

AA

BO B OC

C lrq 1 lrq 2

dflysf] h:t} cfˆgf] sfkLdf b'Oc{ f]6f lrqx¿ lvRg'xf];\ / ∠AOB, ∠BOC j[xt\ sf0] f
∠AOC nfO{ gfKg'xf];\ / tnsf] tflnsfdf egx'{ f];\ .

lrq ∠AOB ∠BOC ∠AOC ∠AOB + ∠BOC + kl/0ffd

∠AOC
1

2

lgisif{ M sg' } laGbs' f] jl/kl/ Ps kl/jm| d0fdf ags] f sf]0fx¿sf] of]ukmn
360° xG' 5 .

ul0ft, sIff & 195

pbfx/0f 1

P R
90°
;u} lbOPsf] lrqaf6 a sf] dfg kQf nufpgx' f;] \ M

;dfwfg (2a)o ao

oxfF ∠POQ + ∠POR + ∠ROS = 180° QO S

cyjf 2a + 90 + a = 180° ⸪ l;wf /v] fdf ags] f cf;Gg
cyjf 3a = 90° sf]0fx¿sf] ofu] kmn 180° xG' 5 .

cyjf a = 30°

pbfx/0f 2

lbOPsf] lrqaf6 x sf] dfg kQf nufpg'xf];\ M

R
Q

40°
S x°

80° O (3x)°

;dfwfg P

oxfF ∠POQ + ∠ROQ + ∠ROS + ∠SOP = 360° ⸪ sg' } laGbd' f jl/kl/
ags] f sf0] fx¿sf]
cyjf 3x + 40° + x + 80° = 360°
of]ukmn 360° xG' 5 .
cyjf 4x + 120° = 360°

cyjf 4x = 360° – 120°

cyjf 4x = 240

cyjf x = 2440
cyjf x = 60°

196 ul0ft, sIff &

pbfx/0f 3

lbOPsf] lrqaf6 ∠POQ / ∠QOR sf] gfk kQf nufpg'xf];\ .

;dfwfg Q

oxfF ∠POQ + ∠QOR = 180° (7x)o
O
cyjf 7x + 3x = 180 (3x)o
R
cyjf 10x = 180 P

cyjf x = 180 = 18
10
ca ∠POQ = 7x = 7 × 18 = 126o

/ ∠QOR = 3x = 3 × 18 = 54o

ctM ∠POQ = 126o / ∠QOR = 54o

cEof; 13.4

1. tn lbOPsf lrqaf6 x, y / b sf] dfg kQf nufpg'xf;] \ . R

-s_ A -v_

70° 2y + 10°

3x + 45° P O b Q
2xo xo


B OC

S

-u_ -3_ B

A B A 80°
7x 50° O
C 7y
3x

O
110°

D

ul0ft, sIff & 197

2. tn lbOPsf tYox¿sf] k/LIf0fåf/f kd| fl0ft ug{x' f];\ .
-s_ b'O{cf6] f l;wf/]vfx¿ Pscfk;df sf6\bf aGg] zLiffl{ ed'v sf]0fx¿
a/fa/ xG' 5g\ .
-v_ l;wf /]vfsf] s'g} laGb'df Psl} t/ ags] f cf;Gg sf0] fx¿sf] of]ukmn 180°
xG' 5 .
-u_ sg' } laGbs' f] jl/kl/ Ps kl/j|md0fdf ags] f sf]0fx¿sf] of]ukmn 360°
x'G5 .

pQ/
1. -s_ 27° -v_ y = 50°, x = 110°, b = 70°
-u_ x = 20° -3_ 40°
2. ;a} kZ| gsf] ;dfwfg sIffdf 5nkmn ugx'{ f;] \ .

198 ul0ft, sIff &

kf7 14 ;dtnLo cfsl[ tx¿

(Plane figures)

14.0 k'g/jnf]sg (Review)

tn lbOPsf lqeh' x¿ -;dafx,' laifdafx', Go"gsf]0fL, ;dsf0] fL, clwsf0] fL s'g
ks| f/sf xg' ,\ sIffdf 5nkmn ugx'{ f;] \ .

-s_ A -v_ Q -u_ M a
4
3 cm 4 cm 4 b
4
B 5.5 cm C P RN a O

-3_ -ª_ -r_

60° 90° 105°
50° 70°

14.1 lqe'hsf] /rgf (Construction of triangle)

14.1.1 bO' c{ f6] f e'hfx¿ / ltgLx¿ larsf] sf]0f lbOPdf lqeh' sf] /rgf
lj|mofsnfk 1

tn lbOPsf r/0fx¿ ckgfO{ PQ = 5.6 cm, QR = 4.5 / ∠PQR = 60° ePsf]

Pp6f lqeh' PQR sf] /rgf ugx{' f;] \ M R

;j{k|yd v];|f lrq lvRgx' f;] \ . 4.5 cm

r/0fx¿

1. PQ = 5.6 cm sf] /]vf v08 lvRgx' f];\ . Q 600 P
5.6 cm
2. laGb' Q df sDkf;sf ;xfotfn] 60° sf] sf]0f

lvRg'xf];\ .

ul0ft, sIff & 199

3. laGb' Q af6 QR = 4.5 cm sf] gfk R4.5 cm
lnP/ sf6g\ x' f;] \ . Q 600

$= laGb' R / P nfO{ hf8] \g'xf];\ . 5.6 cm

ctM cfjZos lqe'h PQR xf] .

P

14.1.2 sg' } Pp6f e'hf / To;df ag]sf b'Oc{ f]6f sf0] fx¿ lbOPdf lqe'hsf] /rgf

lj|mofsnfk 2

tn lbOPsf r/0fx¿ ckgfO{ ∆ABC sf] /rgf ugx'{ f;] \, h;df AB=5.2cm,∠A=75°/
∠B = 60° 5 .

;jk{ y| d v;] f| lrq lvRg'xf];\ . C
r/0fx¿

1. AB = 5.2 cm sf] Pp6f /]vfv08 lvRgx' f];\ .

2. laGb' A df sDkf;sf ;xfotfn] 75° sf] sf0] f lvRg'xf;] \ . A 75° 60° B
3. laGb' B df sDkf;sf ;xfotfn] 60° sf] sf0] f lvRg'xf;] \ . 5.2 cm

4. o;/L 75° / 60° agfPsf /]vfx¿ C
sfl6Psf] laGb'sf] gfd C lbgx' f];\ .

ca cfjZos lqeh' ∆ABC xf] .

A 75° 5.2 cm 60° B

200 ul0ft, sIff &

14.1.3 tLgcf6] } e'hf lbOPdf lqeh' sf] /rgf
ljm| ofsnfk 3

tn lbOPsf r/0fx¿ ckgfO{ AB = 4.5 cm, BC = 5 cm / CA = 6.5 cm ePsf] lqe'h
ABC sf] /rgf ugx{' f;] \ M

;jk{ y| d v;] f| lrq lvRgx' f];\ . C
r/0fx¿

1. AB = 4.5 cm sf] Pp6f /]vf v08 lvRg'xf];\ . 6.5 cm
5 cm
2. laGb' A af6 6.5 cm gfksf] cw{Jof; / laGb' B af6 5 cm
gfksf] cwJ{ of; lngx' f;] \ / Pp6} laGb'df sfl6g] u/L rfk A 4.5 cm B

sf6\g'xf;] \ . C

3. tL bO' c{ f]6f rfk sfl6Psf] laGbs' f] gfd C lbg'xf];\ .

4. /v] f A / C tyf B / C hf]8\gx' f;] \ . 6.5 cm
5 cm
ca cfjZos lqeh' ABC sf] /rgf xf] .

A 4.5 cm B

cEof; 14.1
1. tn lbOPsf cj:yfdf ∆PQR sf] /rgf ugx{' f];\ M

-s_ PQ = 4.8 cm, QR = 5 cm / ∠PQR = 75°
-v_ PR = 5 cm, ∠PRQ = 45° / QR = 5.8 cm
-u_ PQ = 6.2 cm, ∠QPR = 60° / RQ = 6.6 cm
2. tn lbOPsf cj:yfdf ∆ABC sf] /rgf ug'x{ f];\ M
-s_ ∠ABC = 60°, ∠ACB = 45° / BC = 6 cm
-v_ AB = 6.8 cm, ∠BAC = 75° / ∠ABC = 30° cm
-u_ CA = 5.2 cm, ∠ACB = 45° / ∠BAC = 75° cm
3. tn lbOPsf cj:yfdf ∆DEF sf] /rgf ug{'xf];\ M
-s_ DE = 4.5 cm, EF = 4 cm / DF = 5 cm

ul0ft, sIff & 201

-v_ EF = 6.6 cm, DF = 6 cm / DE = 7 cm
-u_ DE = EF = 5.5 cm, DF = 5.2 cm
4. pkoS' t ;ªV\ ofdf ;d"xdf 5nkmn u/L tn lbOPcg;' f/sf efusf] gfk lnP/
lqeh' sf] /rgf ug{sf nflu k|Zgx¿ lgdf0{ f ugx{' f;] \ / /rgf u/L sIffdf k:| t't
ug{'xf];\ .
-s_ bO' {cf6] f eh' fx¿ / ltgLx¿larsf] sf]0fsf] gfk
-v_ tLgcf6] f e'hfx¿sf] nDafOsf] gfk
-u_ Pp6f eh' f / To; eh' fdf ag]sf b'O{cf]6f sf]0fsf] gfk

pQ/
lzIfsnfO{ bv] fpgx' f;] \ .

14.2 ;dfgfGt/ rte'{ 'h, cfot / ju{sf u'0fx¿sf] klxrfg / k/LIf0f
(Identification and verification of the properties of

parallelogram, rectangle and square)

ljm| ofsnfk 1

Pp6f lstfa, Sof/d] af]8{ / 58\s] sf6s] f] sk8fsf] 6'jm| f lng'xf;] \ / tL 7f]; j:ts' f
;txsf cfsf/x¿ s] s:tf 5g\, sIffdf 5nkmn ug'{xf;] \ .

!= lstfa @= Sof/d] af]8{ #= sk8f

tL ;txsf lsgf/f / s'gfdf ags] f sf]0fx¿ gfKgx' f];\ . lsgf/f / sf0] fx¿sf
cfwf/df rt'{e'hsf] u0' faf/] sIffdf 5nkmn ug{x' f];\ .

202 ul0ft, sIff &

14.2.1 ;dfgfGt/ rte{' h' sf u'0fx¿sf] klxrfg (Identification of the

properties of parallelogram)

lj|mofsnfk 2

lbOPsf ;dfgfGt/ rt'e{ h' PQRS sf ;a} e'hfx¿, S R
sf]0fx¿ ljs0fs{ f efux¿ gfKgx' f];\ . ca eh' fx¿sf] Q
;DaGw, sf]0fx¿sf] ;DaGw / ljs0f{sf efux¿sf
;DaGw s] s:tf] bV] ge' of] sIffdf ;fyLx¿;Fu 5nkmn
u/L ;dfgfGt/ rt{e' h' sf u'0fx¿ kQf nufpg'xf];\ . P

k/LIf0f 1

;dfgfGt/ rte' 'h{ sf ;Ddv' sf0] fx¿ a/fa/ xG' 5g\ egL k/LIf0f ug{'xf];\ .

S RPQ

P lrq 1 Q S R

lrq 2

lbOPsf b'Oc{ f6] f ;dfgfGt/ rte' h{' PQRS sf ;a} sf]0fx¿sf gfk lnP/ tnsf]
tflnsfdf eg'{xf];\ .

lrq ∠QPS ∠PQR ∠QRS ∠RSP kl/0ffd

1
2

lgisif{ M ;dfgfGt/ rt{e' 'hsf] ;Ddv' sf]0fx¿ a/fa/ x'G5g\ .

ul0ft, sIff & 203

k/LIf0f 2

;dfgfGt/ rte' h'{ sf ;Ddv' sf]0fx¿ a/fa/ xG' 5g\ egL k/LIf0f ug'{xf];\ .

S R PQ

P Q SR

lrq 1 lrq 2

lbOPsf bO' {cf]6f ;dfgfGt/ rte' h{' PQRS sf ;a} e'hfx¿sf] gfk lnP/ tnsf]
tflnsfdf egx'{ f;] \ .

lrq PQ QR RS SP kl/0ffd

1
2

lgisif{ M ;dfgfGt/ rte{' h' sf] ;Ddv' e'hfx¿ a/fa/ x'G5g\ .

k/LIf0f 3 C

;dfgfGt/ rt'{eh' sf] ljs0f{x¿ ;dlåefhg xG' 5g\ .

A DD
OO

B lrq 1 C AB

lrq 2

lbOPsf b'Oc{ f]6f ;dfgfGt/ rt{e' 'hdf ljs0f{x¿ AC / BD laGb' O df sfl6Psf
5g\ . ca ljs0fs{ f efux¿sf] nDafO gfkL tn lbOPsf] tflnsfdf egx{' f];\ .

lrq AO OC BO OD kl/0ffd

1
2

204 ul0ft, sIff &

lgisif{ M ;dfgfGt/ rt'{eh' sf] ljs0f{x¿ ;dlåefhg x'G5g\ .

;dfgfGt/ rt'eh'{ sf u'0fx¿
-s_ ;dfgfGt/ rte{' h' sf ;Ddv' sf0] fx¿ a/fa/ xG' 5 .
-v_ ;dfgfGt/ rte{' h' sf ;Dd'v e'hfx¿ a/fa/ x'G5 .
-u_ ;dfgfGt/ rt'{e'hsf ljs{0fx¿ k/:k/ ;dlåefhg xG' 5g\ .

14.2.2 cfotsf u'0fx¿sf] klxrfg (Identification of the properties of

Rectangle)

ljm| ofsnfk 3

Pp6f sfkLsf] kfgf lng'xf;] \ / o;sf ;Ddv' lsgf/fx¿ / S R

sg' fdf ePsf sf]0fx¿ / ljs0f{x¿sf nDafO gfKg'xf;] \ .

o;/L gfKbf s] u'0f kQf nfUb5, sIffdf ;fyLx¿;Fu

5nkmn ugx'{ f;] \ . PQ
k/LIf0f 1

cfotsf ;a} sf0] fx¿ 90° sf x'G5g\ . Q
P
SR

R

P Q

lrq 1 lrq 2 S

lbOPsf bO' c{ f6] f cfot PQRS sf kT| o]s sf]0fx¿ gfkL tn lbPsf] tflnsfdf
eg{'xf;] \ .

lrq ∠QPS ∠PQR ∠QRS ∠RSP kl/0ffd

1
2

lgisif{ M cfotsf ;a} sf0] fx¿ 90° sf x'G5g\ .

ul0ft, sIff & 205

k/LIf0f 2

cfotsf ;Ddv' eh' fx¿ a/fa/ x'G5g\ . S

SR

P
R

P lrq 1 Q lrq 2

Q

lbOPsf b'Oc{ f]6f cfot PQRS sf ;a} e'hfx¿ gfkL tn lbOPsf] tflnsfdf
eg'{xf;] \ / ltgLx¿larsf] ;DaGw kQf nufpg'xf];\ .

lrq PQ QR RS SP kl/0ffd

1
2

lgisif{ M cfotsf ;Ddv' e'hfx¿ a/fa/ x'G5g\ .

k/LIf0f 3

cfotsf las0{ fx¿ a/fa/ x'G5g\ . C

AD

D

BC B

lrq 1 lrq 2 A

lbOPsf b'O{cf]6f cfot ABCD sf ljs0fx{ ¿ AC / BD sf gfk lnO{ tn lbOPsf]
tflnsfdf egx{' f];\ .

lrq AC BD kl/0ffd

!

@

lgisif{ M cfotsf las0{ fx¿ a/fa/ x'G5g\ .

206 ul0ft, sIff &

cfotsf u'0fx¿
-s_ cfotsf ;a} sf]0fx¿ 90° sf x'G5g\ .
-v_ cfotsf ;Dd'v e'hfx¿x¿ a/fa/ xG' 5g\ .
-u_ cfotsf ljs0fx{ ¿sf] nDafO a/fa/ x'G5g\ .

14.2.3 ju{sf u0' fx¿sf] klxrfg (Identification of the properties of

Square)

lj|mofsnfk 4

Pp6f ldgL r];sf] af8] { jf jufs{ f/ ;tx ePsf] 7f;] j:t' lng'xf];\ / pSt
r];sf] af8] { jf 7f]; j:tx' f] juf{sf/ ;txnfO{ sfkLdfly /fvL jl/kl/ 3]/f
nufpg'xf;] \ / ljs0f{x¿ hf]8\g'xf;] \ . o;/L ags] f] rte{ 'hsf] ;a} eh' fx¿, sf]0fx¿,
ljs0fs{ f] nDafO, ljs0f{larsf] sf0] f, ljs0fs{ f efux¿ / zLif{sf0] fsf ljeflht
sf0] fx¿ gfKg'xf;] \ . o;/L gfKbf s] s:tf kl/0ffdx¿ kQf nfU5 sIffdf ;fyLx¿;Fu
5nkmn ugx'{ f];\ .

k/LIf0f 1

jus{ f ;a} sf]0f / eh' f a/fa/ x'G5g\ . W

XY X

WZ Z

lrq 1 Y

lrq 2

lbOPsf b'Oc{ f]6f ju{ WXYZ sf ;a} e'hf / sf]0f gfkL tn lbOPsf] tflnsfdf
eg'x{ f];\ .
lrq ∠X ∠Y ∠Z ∠W XY WX YZ ZW kl/0ffd

1

2

lgisif{ M ju{sf ;a} sf]0f 90° / ;a} e'hfsf] nDafO a/fa/ x'G5 .

ul0ft, sIff & 207

k/LIf0f 2 R
ju{sf ljs0fx{ ¿sf] nDafO a/fa/ x'G5g\ . Q

PS
S

Q lrq 1 R P

lrq 2

lbOPsf bO' c{ f]6f ju{ PQRS sf ljs0f{x¿ PQ n RS sf] nDafOsf] gfk lnO{ tnsf]
tflnsfdf eg'{xf];\ .

lrq PR QS kl/0ffd

1
2

lgisif{ M ju{sf ljs0f{x¿sf] nDafO a/fa/ x'G5g\ .

k/LIf0f 3

ju{sf ljs0fx{ ¿ cfk;df nDafw{s xG' 5g\ . R

PS

O

SO Q

Q R lrq 2 P

lrq 1

lbOPsf bO' {cf]6f ju{ PQRS sf ljs0f{sf efux¿ / ljs0f{x¿larsf sf]0fx¿sf] gfk
lnO{ tnsf] tflnsfdf egx{' f];\ .

208 ul0ft, sIff &

lrq PO OR QO OS ∠POQ ∠POS kl/0ffd

1
2

lgisif{ M jus{ f ljs0f{ cfk;df nDafws{ xG' 5g\ .

k/LIf0f 4

ju{sf ljs0fx{ ¿n] zLif{sf]0fx¿nfO{ cfwf ub{5 . R

PS

O S Q
O

Q R

lrq 1 lrq 2 P

lbOPsf b'O{cf]6f ju{ PQRS sf zLifs{ f0] fx¿ / ljs0f{x¿n] ljefhg u/s] f zLif{sf0] fsf
efux¿sf gfk lnO{ tnsf] tflnsfdf egx'{ f;] \ M

lrq 1 ;xfos sf]0fx¿sf] gfk kl/0ffd
zLifs{ f]0fsf] gfk

∠PQR = ............ ∠PQS = .......... / ∠SQR = ..........

∠QRS = ............ ∠QRP = .......... / ∠PRS = ..........

∠RSP = ............ ∠RSQ = .......... / ∠QSP = ..........

∠SPQ = ............ ∠SPR = .......... / ∠RPQ = ..........

ul0ft, sIff & 209

lrq 2 ;xfos sf]0fx¿sf] gfk kl/0ffd
zLif{sf0] fsf] gfk

∠PQR = ............ ∠PQS = .......... / ∠SQR = ..........

∠QRS = ............ ∠QRP = .......... / ∠PRS = ..........

∠RSP = ............ ∠RSQ = .......... / ∠QSP = ..........

∠SPQ = ............ ∠SPR = .......... / ∠RPQ = ..........

lgisif{ M jus{ f kT| os] ljs0f{n] sf0] fnfO{ cfwf ub{5 .

jus{ f u0' fx¿
-s_ jus{ f ;a} sf0] fx¿ / eh' fx¿ cfk;df a/fa/ x'G5g\ .
-v_ ju{sf ljs0fx{ ¿sf] nDafO cfk;df a/fa/ xG' 5g\ .
-u_ jus{ f ljs0fx{ ¿ cfk;df nDaf{ws xG' 5g\ .
-3_ ju{sf ljs0f{x¿n] zLif{ sf]0fx¿nfO{ cfwf ub{5 .

cEof; 14.2

tn lbOPsf egfO l7s 5g\ jf 5g} g\ 5'6\ofpgx' f];\ .
-s_ ;a} rt'eh{' sf ;Dd'v ehfx¿ a/fa/ xG' 5g\ .
-v_ ;dfgfGt/ rt'eh{' sf ;Dd'v sf]0fx¿ a/fa/ x'G5g\ .
-u_ ju{sf ;Dd'v e'hfx¿ dfq a/fa/ x'G5g\ .
-3_ cfotsf ljs0f{x¿ cfk;df nDafw{s x'G5g\ .
-ª_ cfotsf ;a} u'0f ;dfgfGt/ rte' 'h{ df klg x'G5 .
-r_ ju{sf ljs0f{x¿ a/fa/ xG' 5g\ .
-5_ cfotsf ;Ddv' sf]0fx¿ dfq a/fa/ x'G5g\ .

210 ul0ft, sIff &

kl/of]hgf sfo{
cfˆgf] 3/ / ljBfno jl/kl/ /xs] f cfotfsf/, jufs{ f/ / ;dfgfGt/
rt'e{h' cfsf/sf ;txx¿ ePsf j:t'x¿ vf]Hg'xf];\ . pSt j:t'sf
;txx¿ sfkLdf 6«;] u/L cfot, ju{ / ;dfgfGt/ rt'e{'hsf u'0fx¿
k/LIf0f ug'x{ f;] \ / sIffdf k|:tt' ug{x' f;] \ .

pQ/
lzIfsnfO{ bv] fpg'xf];\ .

14.3 kfOyfuf]/; ;fWo (Pythagoras Theorem)

kfOyfuf]/; ;fWosf] k/LIf0f

Pp6f ;dsf0] fL lqe'h lvRgx' f;] \ h;df ∠Q = 90° 5 . P
pSt ;dsf0] fL lqeh' sf] e'hfx¿ gfKg'xf;] \ / k|Tos] eh' fdf

Ps Pscf]6f ju{ agfpgx' f;] \ .

ca ju{ A, B / C sf] Ifq] kmn lgsfNg'\xf;] \ . sIffdf R

;fyLx¿;Fu 5nkmn ug{'xf];\ . s] ju{ A sf] Ifq] kmn = ju{
B sf] If]qkmn + ju{ C sf] If]qkmn xG' 5, 5nkmn ugx'{ f];\ . Q

oxfF s0fd{ f ag]sf jus{ f] If]qkmn cGo b'O{ eh' fdf
ags] f ju{sf] of]ukmn;Fu a/fa/ eof] .

ctM -s0f{_2 = -nDa_2 + -cfwf/_2 PA
B
h2 = p2 + b2
Q R
hxfF h s0f{ (hypotenuse), p nDa C
(perpendicular) / b cfwf/ (base) 5g\ .

ul0ft, sIff & 211

;dsf]0fL lqe'hdf s0f{sf] ju{ c¿ bO' { e'hfsf] ju{sf]
of]ukmn;Fu a/fa/ xG' 5 .

lj|mofsnfk 1

Pp6f ;]6\ :SjfP/ lngx' f];\ / kT| os] lsgf/fsf] nDafO
gfKgx' f];\ . ;a} eGbf nfdf] lsgf/f 5'6\ofpg'xf;] \ . ;aeGbf
nfdf] lsgf/fsf] gfksf] ju{;Fu cGo bO' { lsgf/fsf] gfksf] jus{ f]
of]ukmn a/fa/ x'G5 jf x'bg} sIffdf 5nkmn ug{x' f;] \ / lgisif{
lgsfNg'xf];\ .

pbfx/0f 1

lbOPsf] lqeh' ;dsf]0fL xf] jf xfO] g kQf nufpgx' f;] \ . 12 cm 13 cm
;dfwfg 5 cm
dfgfF} s0f{ (h) = 13 cm
nDa (P) = 12 cm
cfwf/ (b) = 5 cm
ca h2 = p2 + b2
cyjf (13)2 = (12)2 + (5)2
cyjf 169 cm2 = 144 cm2 + 25 cm2
cyjf 169 cm2 = 169 cm2

oxfF s0fs{ f] ju;{ Fu bO' { eh' fsf] jus{ f] of]ukmn a/fa/ eof] . ctM pSt lqe'h
;dsf]0fL lqe'h xf] .

212 ul0ft, sIff &

pbfx/0f 2

lbOPsf] ;dsf0] fL lqe'hdf yfxf gePsf] e'hf kQf nufpgx' f];\ M

;dfwfg

oxf F ∠PQR = 90°

s0f{ (h) = PR = 5 cm

nDa (P) = PQ = ? P

cfwf/ (b) = QR = 4 cm 5 cm
ca h2 = p2 + b2

cyjf 52 = (PQ)2 + 42 Q 4 cm R
cyjf 25 = PQ2 + 16

cyjf 25 – 16 = PQ2

cyjf PQ2 = 9

cyjf PQ = 3 cm

ctM yfxf gePsf] e'hf PQ sf] nDafO 3 cm x'G5 .

cEof; 14.3

1. tn lbOPsf jfSox¿ l7s eP 'T' / a]l7s eP 'F' n]Vg'xf;] \ .
-s_ ;a} lqeh' sf] nfdf] e'hf s0f{ xG' 5 .
-v_ ;dsf]0fL lqehdf dfq kfOyfuf]/; ;fWo dfGo xG' 5 .
-u_ ;dsf0] fL lqe'hdf ;dsf]0f agfpg] Pp6f e'hf s0f{ xg' 'k5{ .
-3_ ;dsf0] fsf] ;Ddv' eh' f lqe'hsf] s0f{ x'G5 .

2. tn lbOPsf gfk ePsf sg' s'g lqeh' ;dsf0] fL lqe'h x'g\ <
-s_ 12 cm, 10 cm, 5 cm -v_ 13 cm, 12 cm, 5 cm
-u_ 15 cm, 16 cm, 17 cm -3_ 8 cm, 15 cm, 17 cm

ul0ft, sIff & 213

3. tn lbOPsf ;dsf]0fL lqe'hdf yfxf gePsf] eh' fsf] nDafO kQf nufpg'xf];\ M

A -v_ M -u_ 5 cm
Q
-s_ R

5 cm 17 cm 12 cm

15 cm PP
B 3 cm C

N

D

-3_ W 8 cm Y -ª_ 12 cm

A 90° ?

4 cm

17cm

X B 3 cm C

kl/of]hgf sfo{

cfˆgf] 3/ / ljBfno jl/kl/ /x]sf lqeh' cfsf/sf ;tx ePsf j:tx' ¿ vfh] L
tL ;txx¿sf lsgf/fsf] nDafOsf gfk lngx' f;] \ / kfOyfuf]/; ;fWo k/LIf0f
ugx{' f;] \ . cfˆgf] sfo{ sIffdf k:| tt' ugx'{ f;] \ .

pQ/ -v_ T -u_ F -3_ T

1. -s_ F -v_ xf ] -u_ xfO] g -3_ xf]
2. -s_ xf]Og
3. -s_ 4 cm -v_ 8 cm -u_ 13 cm -3_ 15 cm -ª_ 13 cm

214 ul0ft, sIff &

kf7 15 cg'¿k cfs[ltx¿

(Congruent Figures)

15.0 kg' /jnfs] g (Review)

tn lbPOsf cfsl[ tx¿dWo] sg' sg' p:t} cfsf/sf / a/fa/ gfksf 5g,\
56' \ofpg'xf];\ M

15.1 cg¿' k cfsl[ tx¿ (Congruent Figures)

ljm| ofsnfk 1

sIffsf ;Dk0" f{ ljBfyL{x¿ rf/ ;d"xdf ljefhg ug{ k|Tos] ljBfyL{n] cfcfˆgf]
Hofldlt afs;df ePsf] ;6] :SjfP/ lnP/ cfˆgf] ;dx" sf] ;fyLsf] ;]6 :SjfP/ dfly
/fVgx' f;] \ -vK6\ofpg'xf];_\ / tn' gf u/L tnsf kZ| gsf] pQ/ vf]Hg'xf;] \ M

vK6o\ fPsf ;]6 :Sjfo/

ul0ft, sIff & 215

-s_ ;6] :Sjfox¿ s:tf cfsf/sf 5g\ <

-v_ vK6\ofPsf ;]6 :Sjfox¿sf gfk ;dfg÷a/fa/ 5g\ ls km/s km/s 5g\ <

-u_ cfsf/ p:t} / gfk klg ;dfg ePsf ;6] \ :Sjfo/x¿nfO{ Ps 7fpFdf /fvL
k|bz{g ug'x{ f];\ / s] oL ;6] \ :Sjfox¿ cfk;df cg¿' k 5g\ < 5nkmn u/L
sIffdf k:| t't ugx'{ f;] \ .

ljm| ofsnfk 2

Ps Pscf]6f cfotfsf/ sfuhsf 6'j|mf lngx' f];\ . lrqdf
bv] fP h:t} u/L l7s lardf k6o\ fpgx' f;] \ . k6o\ fPsf]
sfuhnfO{ vfn] ]/ k6l\ 6Psf] 7fFpdf s}+rLn] sf6g\ x' f;] \ . bj' }
6'jm| fnfO{ vK6\ofpg'xf];\ / t'ngf u/L ;dx" df tnsf k|Zgsf]
pQ/ vf]hL u/L sIffdf k:| t't ugx{' f];\ .
-s_ bj' } cfs[ltx¿ p:t} cfsf/sf 5g\ ls 5}gg\ <
-v_ bj' } 6j' |mfsf gfkx¿ a/fa/ 5g\ ls 5}gg\ <
-u_ cfsf/ p:t} / a/fa/ gfk ePsf cfs[ltx¿nfO{ s:tf cfsl[ tx¿ elgG5 <

ljm| ofsnfk 3

Pp6f a]Grdf a;]sf ;fyLx¿sf] Pp6f ;d"x x'g] u/L ;dx" df ljefhg eO{ tn
lbOPsf k|Tos] hf8] L lrqx¿nfO{ 6]«l;ª kk] /sf] ;xfotfn] sfkLdf agfpgx' f;] \ .

AP M PA D

B CQ RN OB C

cfkmn" ] agfOPsf tL kT| os] hf8] L lrqsf] aflx/L 3]/f srF} Lsf] ;xotfn] sf6\g'xf;] \ .
To;kl5 Pp6f lrqdfly csf]{ lrq vK6o\ fO{ t'ngf ug{'xf];\ / tnsf] kZ| gx¿sf]
pQ/ vf]Hg'xf];\

216 ul0ft, sIff &

-s_ klxnf] hf8] L lrqx¿ -lqe'hx¿_ p:t} cfsf/sf 5g\ ls 5g} g\ <
-v_ klxnf] hf8] L lrqx¿ -lqeh' x¿_ sf gfk a/fa/ 5g\ ls 5}gg\ <
-u_ To:t} bf];|f] hf8] L lrqx¿ -rte' '{hx¿_ df s]s] s/' fx¿df ;dfgtf 5 <
;fyL;uF 5nkmn ugx'{ f];\ / lgisif{ sIffdf k:| tt' ug'{xf;] \ .

ljm| ofsnfk 4

kT| o]s ljBfyLn{ ] Ps Pscf6] f ;]6 :Sjfo/ lnO{ cfcfˆgf] sfkLdf -;]6\ :Sjfo/_
/fv]/ To;sf] aflx/L 3]/f 6;]« u/L b'O{ b'Oc{ f]6f lqe'hx¿ agfpgx' f;] \ . k|To]s lqeh' sf]
gfdfª\sg j|mdzM ABC / XYZ ugx'{ f;] \ .

AX

B CY Z

kT| o]s lrqsf] aflx/L 3]/f sf6]/ lqeh' ABC nfO{ lqeh' XYZ dfly /fvL bfFHgx' f];\ /
tnsf] tflnsf egx{' f;] \ M

lqe'h XYZ sf]
laGb' X dfly lqeh' ABC sf] laGb' =================5 .
laGb' Y dfly laGb' ===================5 .
laGb' Z dfly laGb' ===================5 .
To:t} u/L,
e'hf XY dfly e'hf ===================5 .
eh' f YZ dfly e'hf=====================5 .
eh' f ZX dfly e'hf ==================5 .
XY = .........., YZ = ............. / ZX = ................. 5 .

ul0ft, sIff & 217

lqe'h ABC / lqe'h XYZ nfO{ s:tf lqe'hx¿ eGg ;lsG5 < ;fyL;uF 5nkmn
ug{x' f;] \ / lgisif{ sIffdf k:| tt' ug'{xf];\ M

p:t} cfsf/ / a/fa/ gfk ePsf cfsl[ tx¿nfO{ cg¿' k cfs[ltx¿
(congrnent figures) elgG5 .

pbfx/0f 1

tnsf sg' sg' cfs[ltx¿ cg¿' k 5g,\ lsg < -O{_

-c_ 2 cm -cf_ 4 cm

2 cm

2 cm 2 cm 2 cm 2 cm 3 cm 3 cm

2cm 2 cm 4 cm

;dfwfg

oxfF cfsl[ tx¿ -c_ / -cf_ cg¿' k 5g\ lsgls ltgLx¿sf cfsf/ p:t} 5g\ / eh' fsf
gfkx¿ klg a/fa/ 5g\ .

pbfx/0f 2
tnsf cfs[ltx¿ cjnf]sg u/L sg' sg' cfs[ltx¿ cg¿' k 5g,\ lsg <
-c_ -cf_ -O_

;dfwfg

oxfF cfs[ltx¿ -c_ / -cf_ cg¿' k 5g\ lsgls ltgLx¿sf cfsf/ p:t} / gfk]/
xb] f{ gfk klg a/fa/ kfOof] .

218 ul0ft, sIff &

cEof; 15

1. tnsf s'g sg' cfs[ltx¿ cg¿' k lsg< 2 cm

3 cm

-s_ -c_ 3 cm 3 cm -cf_ 2 cm 2 cm

3 cm 2 cm
6 cm
-v_ -c_ -cf_

6 cm 6 cm 6 cm

6 cm 6 cm

-u_ -c_ 4cm -cf_ 5cm

2cm 2cm 3cm 3cm
4cm 5cm

2. tnsf s'g sg' cfs[ltx¿ cg¿' k 5g\ < -cf_
s_ -c_

-v_ -c_ -cf_

-u_ -c_ -cf_

ul0ft, sIff & 219

-3_ -c_ -cf_

-ª_ -c_ -cf_

-r_ -c_ -cf_

-5_ -c_ -cf_

3. kfrF kfFrcf]6f km/s km/s 7f;] j:t' ko| f]u u/L cg¿' k cfsl[ t lvRgx' f;] \ .
4. cfˆgf] b'j} xft hf]8/] gd:sf/ ug'{xf;] \ . tL b'j} xTs]nfx¿ Ps csf{;Fu cg'¿k

5 ls 5g} < ;fyL;uF 5nkmn ug'x{ f;] \ .
kl/of]hgf sfo{

;a} ljBfyL{x¿ pkoS' t ;d"xdf ljefhg eO{ kT| os] ;dx" n] cfcfˆgf] 3/
tyf ljBfno j/k/ ePsf cg'¿k cfsl[ tx¿ h:t}M l;Ssf, gf]6, ?dfn,
lstfa, O/]h/ cflb ;ª\sng u/L sIffdf kb| z{ g ug{x' f];\ .

pQ/
lzIfsnfO{ bv] fpgx' f;] \ .

220 ul0ft, sIff &

kf7 16 7f;] j:tx' ¿

(Solid Objects)

15.0 k'g/jnf]sg (Review)

aG] rdf ;Fu} a;s] f ;fyLx¿;Fu 5nkmn u/L tnsf] tflnsf egx'{ f];\ M

j:t'sf] gfd ;dtnLo cfsl[ tx¿ 7f;] cfs[ltx¿
;nfO{sf] a66\ f cfot if8\dv' f
8fO; ju{ 3g
cfO;ljm| dsf] aflx/L vf]n jQ[
8«d

dflysf] tflnsfdf ePsf 7f]; cfsl[ t / ;dtnLo cfsl[ tsf af/d] f ;fyLx¿;uF
5nkmn ug{'xf;] / lgisif{ sIffdf k:| t't ug'{xf];\ .

16.1 66] f« x8] g« (Tetrahedron)
lj|mofsnfk 1

pko'St ;ªV\ ofdf ljBfyLx{ ¿sf] ;d"x agfpg'xf];\ / kT| os]
;d"xn] Ps Pscf]6f bfofFsf] lrqdf ePsf] h:t} 7f]; j:t'
lngx' f;] \ . tL j:t'sf] cjnfs] g u/L tnsf k|Zgx¿sf
af/]df ;d"xdf 5nkmn ugx'{ f;] \ M

-s_ s] o;sf ;a} lsgf/fx¿ a/fa/ 5g\ <
-v_ s] k|Tos] ;tx ;dafx' lqe'h cfsf/sf 5g\ <
-u_ o;sf ;txx¿ sltcf]6f 5g\ <
-3_ o;df slt sltcf6] f lsgf/f / zLif{ laGbx' ¿ 5g\ <
-ª_ of] lgoldt 7f;] j:t' jf clgoldt 7f]; j:t' s'g xf] <
;d"xdf 5nkmn u/L ;s]kl5 ;d"xsf] lgisifn{ fO{ sIffdf k:| t't ugx{' f];\ .

66] «fx8] «g Pp6f lgoldt HofldtLo 7f;] cfsl[ t xf] . o;sf k|To]s ;txx¿

;dafx' lqeh' af6 ags] f xG' 5g\ . o;df hDdf 4 cf]6f ;txx¿ 4 cf6] f
zLif{laGb' / 6 cf6] f lsgf/fx¿

ul0ft, sIff & 221

16.1.1 6]6«fx]8«gsf] vf]j|mf] gd'gf (Skeleton of Tetrahedron) lgdf{0f
ljm| ofsnfk 2

pko'St ;ªV\ ofdf ljBfyL{x¿sf] ;dx" agfpgx' f;] \ . k|To]s
;d"xn] 6 cf6] f a/fa/ gfksf l;Gsfx¿ / 4 6'jm| f cfn' jf
cGo g/d j:t'sf 6'j|mfx¿ lng'xf;] \ . ca lrqdf bv] fP
h:t} u/L l;Gsfx¿ / cfns' f 6'j|mfx¿ hf8] g\ x' f]; . To;kl5
cjnf]sg u/L ;d"xdf ;fyLx¿;Fu 5nkmn ug'x{ f;] \ / tnsf
k|Zgx¿sf] pQ/ vf]Hgx' f;] \ / sIffdf k:| tt' ug'x{ f];\ .

-s_ s:tf] cfsl[ t aGof] <
-v_ sltcf]6f lsgf/fx¿ / sltcf]6f sg' fx¿ ag] <

16.2 cS6fx]8«g (Octahedron)

lj|mofsnfk 3
ljBfyLx{ ¿sf] ;dx" agfpgx' f;] \ / k|Tos] ;d"xn] Ps
Pscf6] f bfofFsf] lrqdf ePsf] h:t} 7f;] j:t' lng'xf];\
tL 7f]; j:ts' f] cjnf]sg u/L tnsf kZ| gx¿sf af/]df
;d"xdf 5nkmn ugx{' f];\ .
-s_ s] o;sf k|Tos] lsgf/fx¿ a/fa/ 5g\ <
-v_ s] o;sf kT| os] ;tx ;dafx' lqe'h cfsf/sf 5g\ <
-u_ o;sf sltcf]6f ;txx¿ 5g\ <
-3_ o;df sltcf]6f lsgf/fx¿ / cltcf]6f zLif{ laGbx' ¿ 5g\ <
-ª_ of] 7f;] cfs[ltsf] gfd s] xf] <
-r_ of] lgoldt 7f]; j:t' jf clgoldt 7f;] j:t' s'g xf] <

cS6fx8] g« Pp6f lgoldt 7f]; j:t' xf] . o;sf k|To]s ;txx¿ ;dafx'
lqe'h cfsf/sf x'G5g\ . o;df hDdf 8 cf6] f ;txx¿, 6 cf]6f zLif{
laGbx' ¿ / 12 cf6] f lsgf/fx¿ x'G5g\ .

222 ul0ft, sIff &

16.2.1 cS6fx8] «gsf] vfj] m| f gdg' f (Skeleton of octahedron ) lgdf0{ f

lj|mofsnfk 4

pko'St ;ª\Vofdf ljBfyL{sf] ;dx" agfpgxf;] \ .
k|Tos] ;d"xn] 12 cf]6f a/fa/ gfksf l;Gsfx¿ / 6
cf]6f cfns' f 6'jm| fx¿ jf cfn' h:t} j:t'sf 6j' |mfx¿
lngx' f];\ ca lrqdf bv] fP h:t} u/L l;Gsfx¿ /
cfn'sf 6j' |mfx¿ hf8] \g'xf;] \ . To;kl5 cjnf]sg
u/L ;dx" df 5nkmn u/L tnsf kZ| gx¿sf] pQ/
vfH] g'xf];\ / sIffdf k:| t't ugx{' f];\ .

-s_ of] cfsl[ t -7f;] cfsl[ t_ sf] gfd
s] xf] <

-v_ o;sf sltcf6] f lsgf/fx¿ 5g\ <
-u_ o;df sltcf]6f zLifl{ aGb'x¿ 5g\ <
-3_ o;sf ;txx¿ sltcf]6f / s:tf s:tf 5g\ <

ljm| ofsnfk 5

sIffsf ;Dk0" f{ ljBfyLx{ ¿ kfrF cf]6f ;d"xdf ljeflht eP/ a:g'xf];\ .

kT| o]s ;d"xdf j|mdzM 3g (cube), 66] f« x]8g« (Tetrahedron),cS6fx]8g« (Octahedron),
8f]8]sfx]8g« (Dodecahedron) / cfOsf];fx8] «g (Icosahedron) sf 7f;] gdg" fx¿
cjnfs] g u/L tnsf kZ| gx¿sf] af/]df 5nkmn ug{x' f];\ .

-s_ lbOPsf] 7f;] cfs[ltdf sltcf6] f lsgf/fx¿ 5g\ u0fgf ug{'xf];\ .
-v_ lbOPsf] 7f]; cfsl[ tdf sltcf6] f ;dtn ;txx¿ 5g\, u0fgf ug{x' f;] \ .
-u_ lbOPsf] 7f]; cfs[ltdf sltcf]6f zLif{ laGb'x¿ 5g\, u0fgf ug{x' f;] \ .

ca ;a} ;dx" n] cfˆgf] ;d"xn] u0fgf u/sf sg' fx¿sf] ;ª\Vof, ;txx¿sf] ;ªV\ of
/ lsgf/fx¿sf] ;ªV\ of tnsf] tflnsfdf eg'{xf;] . To;kl5 ltgLx¿sf] ;DaGwsf
af/]df 5nkmn u/L sIffdf k|:t't ugx'{ f];\ .

ul0ft, sIff & 223

j|m=;= 7f;] j:tx' ¿ zLif{ lsgf/fx¿sf] ;txx¿sf] V, E / F sf]
laGb'x¿sf] ;ª\Vof (E) ;ª\Vof (F) ;DaGw
1 3g ;ª\Vof (V)
2 6]6«fx]8g« V–E+F=
3. cS6fx]8«g
4. 8f8] ]sfx8] «g
5. cfOsf;] fx8] «g

3g, 66] «fx]8«g, cS6fx]8g« , 8f]8]sfx]8g« / cfOsf;] fx]8g« df zLif{ laGb'x¿sf]
;ªV\ of (V) ;txx¿sf] ;ª\Vof (F) / lsgf/fx¿sf] ;ªV\ of (E) sf] ;DaGw
V – E + F = 2 x'G5 .

pbfx/0f 1

Pp6f 6]6f« x8] «gdf 4 cf6] f zLiflaGbx' ¿ / 6 cf]6f lsgf/fx¿ 5g\ eg] ;txx¿sf]
;ªV\ of kQf nufpg'xf;] \ .
;dfwfg
oxfF 6]6f« x]8g« sf zLiflaGb'sf] ;ª\Vof (V) = 4
lsgf/fsf] ;ª\Vof (E) = 6
;txsf] ;ª\Vof (F) = ?
xfdLnfO{ yfxf 5,

V–E+F=2

cyjf 4 – 6 + F = 2
cyjf –2 + F = 2
cyjf F = 2 + 2

F=4

224 ul0ft, sIff &

cEof; 16.1

1. tnsf jfSox¿ l7s jf a]l7s s] xg' , 56' o\ fpgx' f];\ M

-s_ 66] f« x8] «gsf kT| o]s ;tx ;dafx' lqeh' cfsf/sf xG' 5g\ .
-v_ 66] f« x8] «gsf ;a} lsgf/fx¿ a/fa/ x'G5g\ .
-u_ 66] «fx]8«gdf hDdf tLgcf]6f ;txx¿ x'G5g\ .
-3_ cS6fx]8«gsf kT| o]s ;tx ;dafx' lqe'h cfsf/sf x'Fbg} g\ .
-ª_ cS6fx8] «gdf hDdf 4 cf]6f lsgf/fx¿ xG' 5g\ .
2. tnsf kZ| gx¿sf] pQ/ nV] g'xf];\ M

-s_ 66] «fx8] g« eg]sf] s] xf] <
-v_ 66] f« x8] «g / cS6fx8] «gsf s'g} bO' c{ f]6f km/s nV] gx' f;] \ .
-u_ 8f]8s] fx]8g« / cfOsf;] fx]8g« sf ;tx, lsgf/f tyf sg' fsf] ;DaGw

hgfpg] ;q" nV] g'xf];\ .
-3_ cS6fx]8g« sf] k|Tos] ;tx (Face) s:tf] cfsf/sf] x'G5 <
-ª_ cS6fx8] «gdf sltcf6] f zLiflaGb'x¿ (Vertices) / lsgf/fx¿ (Edges)

x'G5g\ <
3. Pp6f 66] f« x]8g« df lsgf/fx¿ (Edges) / ;txx¿ (Surfaces) sf] ;ªV\ of

j|mdzM 6 / 4 5 eg] sg' f zLif{x¿ (Vertices) sf] ;ª\Vof kQf nufpg'xf;] \ .
4. Pp6f cS6fx8] «gdf ;txsf] ;ªV\ of slt ePdf pSt cS6fx]8g« sf] s'gfx¿sf]

;ª\Vof 6 x'G5, kQf nufpgx' f;] \ .

kl/of]hgf sfo{

h'; vfg] kfOk, 5j\ fnLx¿, afF; tyf lgufnfx¿, 86kg] sf vfnL l/lkmnx¿
tyf wfuf] ko| f]u u//] ljleGg gfksf 3g, 6]6«fx]8«g / cS6fx]8g« sf gd'gfx¿
lgdf{0f u/L sIffdf 5nkmn u/L k|bz{g ug{x' f;] \ .

pQ/
lzIfsnfO{ bv] fpgx' f;] \ .

ul0ft, sIff & 225

16.3 ;fn] L / a]ngf (Cone and Cylinder)

16.3.1 ;fn] L (Cone)
lj|mofsnfk 1

pkoS' t ;ªV\ ofdf ;d"xdf a:gx' f];\ . ;a} ;d"xn] tn lbOPsf h:t} cfsf/sf Ps
Pscf6] f j:tx' ¿ lng'xf];\ . pSt j:t'x¿sf] cjnfs] g u/L tnsf k|Zgx¿sf] pQ/
;dx" 5nkmn u/L vfH] g'xf;] \ M

-s_ k|Tos] j:t' s:tf cfsf/sf 5g\ <
-v_ k|Tos] j:t'sf cfwf/x¿ s:tf cfsf/sf 5g\ <
-u_ kT| o]s j:tx' f] ;tx s:tf] cfsf/sf] 5 <
-3_ kT| os] j:td' f sltcf]6f zLifl{ aGb'x¿ jf s'gfx¿ 5g\ u0fgf ug'x{ f];\ .
dflysf j:tx' ¿df Pp6f sg' f jf zLifl{ aGb', Pp6f cfwf/ jQ[ fsf/ / Pp6f ajm| ;tx
/x]sf 5g\ . oL 7f]; cfsl[ tx¿ ;a} ;f]nL (Cone) xg' \ .

;f]nLsf u'0fx¿
Pp6f zLif{ laGb', Pp6f aj|m ;tx / Pp6f jQ[ fsf/ cfwf/
ePsf] 7f;] cfs[ltnfO{ ;f]nL (Cone) elgG5 .
-s_ Pp6f zLif{laGb' xG' 5 .
-v_ cfwf/ j[Qfsf/ x'G5 .
-u_ Pp6f jjm| ;tx /x]sf] xG' 5 .

226 ul0ft, sIff &

;fn] Lsf vfj] |mf gdg' fsf] (Skeleton Model of Cone) lgdf{0f

lj|mofsnfk 2

3/df k|ofu] ug{] df5f dfg{] 9l8of h:tf ;fn] Lsf vf]jm| f gd'gfx¿sf] ;r" L tof/ u/L
sIffdf ;fyLx¿;uF 5nkmn ugx'{ f];\ .

pkoS' t ;ªV\ ofdf ljBfyL{x¿sf] ;d"x agfpg'xf];\ / kT| o]s ;d"xn]
Ps Pscf6] f j[Qfsf/ 7f;] j:t' / Pp6f sfuh lngx' f];\ . kT| os]
;d"xn] sfuhdfly jQ[ fsf/ 7f]; j:t' /fvL jQ[ agfpgx' f;] \ .
sr}+ Lsf] ;xotfn] j[Qsf] aflx/L 3/] f sf6g\ 'xf];\ . To;kl5 jQ[ nfO{
l7s laraf6 b'O{ k6s k6\ofpg'xf];\ .

A O
OB A
B

ca k6o\ fOPsf] sfuhnfO{ vf]n]/ rf/ efudWo] Ps efu srF} Ln] sf6/] x6fpgx' f];\ /
afsF L /xs] f efunfO{ lrqdf h:t} u/L hf8] /] udn] 6f:F g' xf] . s:tf] cfsl[ t aGof] <
o;df sltcf6] f zLif{laGb', sltcf6] f s'gf / sltcf]6f j[Qfsf/ ;tx 5g\, cjnfs] g
u/L ;d"xsf 5nkmn sIffdf k|bz{g ug'{xf];\ .

ul0ft, sIff & 227

a]ngf (Cylinder )
lj|mofsnfk 3

k|To]s ;dx" ¿n] tn lbOPsf h:t} 7f;] j:t'x¿ lng'xf];\ pSt j:tx' ¿sf÷cfs[ltx¿sf]
cjnf]sg u/L tnsf kZ| gsf] pQ/ ;d"xdf 5nkmn u/L vfH] g'xf;] \ M

-s_ j:tx' ¿ s:tf cfsf/sf 5g\ <
-v_ oL j:tx' ¿sf cfwf/ s:tf cfsf/sf 5g\ <
-u_ s] oL j:t'x¿nfO{ u'8fpg ;lsG5 <
-3_ oL j:tx' ¿df sltcf]6f / s:tf ;dtnLo ;txx¿ 5g,\ u0fgf ug'x{ f;] \ .
oxfF ;a} 7f;] cfs[ltx¿df÷j:tx' ¿df bO' c{ f6] f jQ[ fsf/ ;txx¿ 5g\ . oL ;a}
an] gfsf/ j:t' x'g\ .
a]ngfsf u0' fx¿
-s_ o;sf] cfwf/x¿ j[Qfsf/ x'G5g\ .
-v_ o;df Pp6f jj|m ;tx xG' 5 .
-u_ o;sf cfwf/x¿ cfk;df ;dfgfGt/ x'G5g\ .

cfwf/x¿ j[Qfsf/ / ;dfgfGt/ eO{ Pp6f jj|m ;tx ePsf 7f;] j:tn' fO{
an] gf elgG5 . cyjf a]ngf Pp6f 7f;] j:t' xf], h;sf cfwf/x¿ j[Qfsf/ /
;dfgfGt/ tyf Pp6f jj|m ;tx xG' 5 .

16.3.2 a]ngfsf vf]j|mf gdg' f (Skeleton Model of Cylinder) lgdf{0f
lj|mofsnfk 4

3/, ljBfno, ;8s lsgf/df cfkm"n] b]vs] f bfofsF f] lrqdf ePsf] h:t}
a]ngfsf vfj] m| f gd'gfx¿sf ;"rL tof/ u/L 5nkmn ug{'xf;] \ .
k|To]s ;dx" n] Ps Pscf6] f cfotfsf/ sf8a{ f]8{ k]k/sf] lng'xf];\ . lrqdf
h:t} u/L cfotfsf/ nDafO;Fu a/fa/ kl/lw ePsf pq} b'O{cf6] f j[Qx¿

228 ul0ft, sIff &

lngx' f];\ . lrqdf h:t} u/L cfotfsf/ sf8{af8] { kk] /nfO{ pq} b'Oc{ f]6f jQ[ x¿sf]
kl/lwdf kg{] u/L ag] '{xf;] \ .

A BA BA BA B

C DC DC DC D

To;kl5 sfuhsf wf/x¿nfO{ cfk;df l;wf kg{] u/L udn] 6f:F g'xf;] \ .
s:tf] cfsf/ aGof] <
ags] f] cfsl[ tsf] gfd s] xf] < o;df sltcf]6f jQ[ fsf/ ;txx¿ / sltcf]6f
zLifl{ aGb'x¿ 5g\ < cjnfs] g u/L ;d"xdf 5nkmn ug{'xf];\ / sIffdf k|:t't ug'x{ f;] \ .

cEof; 16.2
1. tnsf k|Zgx¿sf] pQ/ lbg'xf;] \ M

-s_ ;f]nL eg]sf] s] xf] < sg' } bO' c{ f6] f u'0fx¿ n]Vg'xf];\ .
-v_ an] gf eg]sf] s] xf] < sg' } bO' c{ f]6f u'0fx¿ n]Vgx' f];\ .
-u_ a]ngf / ;fn] Ldf ePsf Pp6f ;dfgtf / Pp6f km/s nV] gx' f;] \ .
-3_ an] gf / ;f]nLsf] Pp6f Pp6f gdg' f lrq agfpgx' f;] \ .
2. ;f]nLsf] ;tx / cfwf/ s:tf s:tf x'G5g,\ nV] g'xf];\ .
3. sg' } kfrF cf]6f a]ngfsf/ j:tx' ¿ ;ªs\ ng ug'{xf;] \ . ltgLx¿sf ;tx / cfwf/
s:tf s:tf x'G5g,\ n]Vgx' f;] \ .

kl/ofh] gf sfo{

tkfOs{F f] 3/df ePsf jf 3/df ko| f]u ug]{ kfrF kfFrcf6] f a]ngfsf/ /
;fn] L cfsf/sf j:t'x¿ vfh] L u/L sIffdf k:| t't ugx'{ f];\ .

pQ/
lzIfsnfO{ b]vfpg'xf;] \ .

ul0ft, sIff & 229

kf7 17 lgb{z] fªs\

(Coordinates)

17.0 k'g/jnfs] g

;fyLx¿;Fu ldn]/ lrqdf bv] fP h:t} u/L sfuhsf 6'j|mfx¿df X - cIf / Y- cIfsf
;ªV\ of /v] f agfpg'xf;] \ .

ca bO' { ;d"x (A / B) df afFl8P/ lgb]{zfªs\ v]n v]Ngx' f;] \ .
vN] g] tl/sf

-s_ ;j{ky| d ;ªV\ of kQLx¿nfO{ lrqdf b]vfP h:t} cfk;df nDa xg' ] u/L rp/df
/fVgx' f];\ . ;a} hgf b'O{cf6] f ;d"xdf ljefhg eO{ cfd'Gg] ;fd'Gg] x'g] u/L
a:g'xf;] \ .

+5
+4
+3
+2
+1
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
-1
-2
-3
-4
-5

-v_ klxnf ;d"x A sf ;fyLn] ;dx" B sf ;fyLnfO{ s'g} laGbs' f] lgb{z] fªs\ sf
cfwf/df pSt laGbd' f pleg eGgx' f];\ . l7s :yfgdf pleg g;sd] f aflxl/g'
k5{ eGg] hfgsf/L u/fpgx' f;] \ .

-u_ ca, ;dx" B sf ;fyLn] ;dx" A sf ;fyLnfO{ sg' } laGbs' f] lgbz{] fª\ssf cfwf/df
pSt laGb'df pleg eGg'xf;] \ . n]vflrqsf] cfˆgf] lgb]{zfªs\ df plePsf] ;fyLn]

230 ul0ft, sIff &

g} ;d"x A sf] csf{] ;fyLnfO{ g= -v_ df h:t} u/L sg' } csf{] gofF laGb'sf]
lgbz]{ fª\s eGg] 5 / ;fx] Lcg';f/ pleg'k5{ .

-3 of] lj|mofsnfk dfly h:t} u/L ;a} ;fyLx¿nfO{ Ps Ps k6s s'g} laGb'df
pleg kfpg] u/L cj;/ lbg'kg{] 5 .

-ª_ b'j} ;d"xx¿dWo] h'g ;d"xsf w/} ;fyLx¿ lgbz]{ fª\scg';f/ l7s laGb'df
pleg ;s] pxL ;dx" sf] lht xG' 5 .

17.1 nv] flrqdf laGbs' f] lgbz]{ fªs\

lj|mofsnfk 1

pkoS' t ;ª\Vofdf ljBfyLx{ ¿sf] ;d"x Y A
agfpg'xf];\ / ;uF s} f] u|fkm ePsf] lrqsf] C
cWoog u/L tnsf k|Zgx¿sf] pQ/ X
vfH] gx' f;] \ M B

-s_ XOX' nfO{ s] elgG5 <

-v_ YOY' nfO{ s] elgG5 < X' O
Y'
-u_ laGb' O af6 laGb' A df k'Ug slt

PsfO bfofF uP/ slt PsfO dfly

hfgk' 5{ < D

-3_ laGb' O af6 laGb' D df k'Ug slt
PsfO afofF uP/ slt PsfO tn

hfg'k5{ <

-ª_ laGb' O, B, C / D sf lgbz{] fªs\ s] s] xg\' \ <

dflysf kZ| gx¿sf] pQ/ ;d"xdf 5nkmn u/L sIffdf k|:tt' ug{x' f;] \ .

ljm| ofsnfk 2

;dx" df a;L ;uF }sf] lrqdf 5nkmn u/L tnsf k|Zgx¿sf] pQ/ vfH] gx' f];\ / sIffdf
k|:tt' ugx{' f;] \ M

-s_ laGb' A s'g rt'yfzF{ df k5{ <

ul0ft, sIff & 231

-v_ laGb' A sf] X– lgb]{zfª\s slt Y A
xf]nf < X
B
-u_ To:t} laGb' A sf] Y – lgbz{] fªs\ O C
slt xfn] f <

-3_ laGb' A sf] lgb{z] fªs\ slt xfn] f <

-ª_ laGb' B sf] lgb{]zfªs\ slt xf]nf < X'

-r_ ca laGb' C sf] lgb]{zfªs\ slt
xf]nf <

Y'

sg' } klg laGbs' f] lgbz]{ fª\sn] To; laGbs' f] cjl:yltnfO{ hgfpFbf

X – lgb{]zfª\sn] pbu\ d laGbe' Gbf slt PsfO bfofF jf afofF eGg] a'emfp5F .
To:t} Y – lgbz{] fªs\ n] pb\ud laGbe' Gbf slt PsfO dfly jf tn eGg]
ae' mfpF5 .

17.2 nv] flrqdf lbOPsf laGb'x¿sf] cª\sg (Plotting the Given Points

in the Rraph)

lj|mofsnfk 3 Y A (6, 5)
B (–3, 5)

laGb'x¿ A(6, 5), B (–3, 5),
C (–4, –3) / D(4, –4) nfO{
n]vflrqdf cªs\ g ugx{' f;] \ .

;jk{ |yd sIffsf ljBfyLx{ ¿ X' O X
;d"xdf ljefhg eO{ cf cfˆgf Y' D (4, –4)

u|fkm sfkLdf ;Fus} f] lrqdf

b]vfP h:t} u/L X – cIf / C (–4, –3)

Y – cIf hgfpg] ;ª\Vof /v] fx¿

agfpgx' f];\ .

232 ul0ft, sIff &

-s_ A(6, 5) nfO{ n]vflrqdf cª\sg ug{ s] ug{k' nf{, 5nkmn ug{'xf];\ .

A df kU' g, X – cIfdf pbu\ d laGb' bl] v 6 PsfO bfofF hfgk' g{] xG' 5 . To;kl5
ToxL laGba' f6 Y– cIfdf 5 PsfO dfly uO{ A(6, 5) cªs\ g ug'k{ 5{ .

-v_ To;} u/L j|mdzM laGb' B (-3, 5), C (–4, –3) / D(4, –4) nfO{ s;/L n]vflrqdf
cªs\ g ug{ ;lsG5 xf]nf < ;fyL;uF klg 5nkmn ug{x' f;] \ .

-u_ laGb' A / B larsf] b/' L slt xG' 5 < laGb' A b]lv laGb' B ;Dd ju{ sf7] f ulGt
u//] kQf nufpgx' f;] \ .

pbfx/0f 1

laGbx' ¿ K (3, 4), L(–3, 4), M(3, –5) / N Pp6f cfotsf zLif{ laGb'x¿ xg' \ eg],
-s_ lbOPsf] laGbx' ¿nfO{ n]vflrqdf cªs\ g ug'x{ f;] \ .
-v_ laGb' N sf] lgb]{zfªs\ kQf nufpg'xf];\ .
-u_ laGb' K / L ljrsf b'/L kQf nufpgx' f];\ .
;dfwfg

-s_ lbOPsf] laGb'x¿ K (3,4), L(-3,4), Y K (3, 4)
M(3,-5) nfO{ jm| dzM cªs\ g u/L L (–3, 4) X
;Fus} f] n]vflrqdf b]vfOPsf] 5 .
O M (3, –5)
-v_ laGb' N df k'Ug X- cIfdf pb\ud
laGb'af6 3 PsfO afofF uO{ ToxLaf6 N
5 PsfO tn hfgk' b{5 . t;y{ N Y'
sf] lgb]{zfªs\ (–3, –5) x'G5 . X'

-u_ laGb' K bl] v laGb' L larsf] sf7] f
uGtL ubf{ 6 PsfO 5 .

t;y{ K / L aLrsf] b/' L (KL) = 6
PsfO

ul0ft, sIff & 233

cEof; 17

1. lrqdf lbOPsf HofldtLo cfsl[ tx¿sf zLif{ laGbx' ¿sf lgb]z{ fª\sx¿ kQf
nufpgx' f;] \ .

Y D
PA

Q RB C

X' O P X
J M Q

KL S R

Y'

2. nv] flrqdf lbOPsf cfs[ltx¿sf] zLifl{ aGb'x¿sf] lgbz{] fª\s n]Vg'xf;] \ .

3. tn lbOPsf kT| o]s laGb'nfO{ nv] flrqdf cª\sg ug{x' f];\ .

P(2, 2), Q(–3, 4), R(–2 ,0), S(4, –4), T(-5, –5)

4. tnsf kT| os] laGb'x¿nfO{ nv] flrq agfO cªsg ug{'xf;] \ . k|To]s laGb'nfO{ jm| dzM
hf8] \g'xf];\ . o;/L hf]8b\ f aGg] cfsl[ tsf] gfd klg nV] gx' f];\ .

-s_ A(4, 0), B(4, 4), C(–2, 4) / D (–2, 0)
-v_ R(2, 3), S(2, –2) / T(–1, 2)
5. laGb' A(–2, 3) , B(2, 3), C(–2, 4), D Pp6f cfotsf zLif{ laGbx' ¿ xg' \ eg]
-s_ lbOPsf laGb'x¿nfO{ nv] flrqdf cªs\ g ugx'{ f];\ .
-v_ AB sf] nDafO slt xfn] f <
-u_ CD sf] nDafO slt xf]nf <

234 ul0ft, sIff &

6. laGbx' ¿ J(–4, 4), K (4, 4), L(4, –4) / M Pp6f ju{sf zLif{laGb'x¿ x'g\ eg],
-s_ lbOPsf] laGbx' ¿nfO{ nv] flrqdf cª\sg ug'x{ f;] \ .
-v_ laGb' M sf] lgb]z{ fªs\ n]Vgx' f];\ .
-u_ laGb' JK sf] nDafO slt xf]nf <

kl/ofh] gf sfo{
Pp6f 7'nf] ;fOhsf] u|fkmkk] /df X – cIf / Y – agfpgx' f];\ . pSt
uf| kmdf Pp6f lrq agfpgx' f];\ . pSt lrqsf] cjnf]sg u/L sg' }
kfFrcf6] f laGb'x¿sf] lgbz{] fªsx¿ nV] g'xf];\ / sIffdf k:| t't ugx{' f;] \ .

pQ/
lzIfsnfO{ bv] fpgx' f];\ .

ul0ft, sIff & 235

kf7 18 ;dldlt / 6];n] ;] g

(Symmetry and Tessellation)

18.0 kg' /jnf]sg (Review)

tn lbOPsf lrqx¿sf] cjnfs] g u/L ;dx" df ;fyLx¿;Fu 5nkmn u/L ;f]lwPsf
kZ| gx¿sf] pQ/ vf]Hgx' f;] \ M

-s_ dfly lbOPsf s'g s'g lrqnfO{ bO' { a/fa/ efudf af8F g\ ;lsG5 <
-v_ dfly lbOPsf lrqx¿dWo] sg' sg' ;dldtLo lrqx¿ (Symmetrical

figures) x'g\ 5'6o\ fpgx' f];\ .
-u_ s] dflysf lrqx¿nfO{ 180° sf0] fdf 3d' fpFbf klg p:t} bl] vG5g\ <

● a/fa/ efudf afF8g\ ;lsg] lrqnfO{ ;dldtLo lrq elgG5 .
● sg' } klg lrqdf hg' /v] faf6 lrqnfO{ bO' { a/fa/ efudf k6\ofpg

;lsG5 To; /v] fnfO{ ;dldltsf] cIf elgG5 . o:tf ;dldltsf cIf
PseGbf a9L klg xg' ;S5g\ .

18.1 /v] f / laGb' ;dldlt (Line and Point Symmetry)

18.1.1. /v] f ;dldlt (Line Symmetry)
lj|mofsnfk 1

;a} ljBfyL{n] lbOPsf cfs[ltx¿sf] 6«]; ugx'{ f];\ M

lrq I lrq II lrq III

236 ul0ft, sIff &

cfkmn" ] 6;«] u/s] f lrqnfO{ 86 /]vf -;dldltsf] cIf_ b]vL a/fa/ efudf k6o\ fpg'xf;] \ .
-s_ lrq I nfO{ slt tl/sfn] b'O{ a/fa/ efu xg' ] u/L k6\ofpg ;lsof] <
-v_ lrq II nfO{ slt tl/sfn] a/fa/ efu xg' ] u/L k6o\ fpg ;lsof] <
-u_ lrq III nfO{ slt tl/sfn] a/fa/ efu xg' ] u/L k6o\ fpg ;lsof] <

cfˆgf] aG] rsf ;fyLx¿;uF 5nkmn ug{'xf;] \ .
● lrq I / lrq II nfO{ 1 tl/sfn] k6o\ fpg ;lsG5 . t;y{ o;df /]vLo
;dldltsf] cIf Pp6f dfq 5 .
● lrq III nfO{ 2 tl/sfn] k6o\ fpg ;lsG5 . t;y{ o;df /v] Lo ;dldltsf]
cIf 2 cf6] f 5g\ .

lj|mofsnfk 2
k|To]s ljBfyL{sf] Pp6f sfuhsf] kfgf lnO{ To;nfO{ lar efuaf6 lrqdf
bv] fP h:t} u/L k6\ofpgx' f;] \ . k6\ofOPsf] efunfO{ oyfjt\ /fvL kg' M csf]{ lt/af6
Psk6s k6o\ fpg'xf];\ .

(i) (ii) (iii) (iv) (v)

ca, t;] f]{ lrqdf lbOPsf] cfsl[ t agfpgx' f;] \ . To;kl5 (iv) df h:t} u/L To;sf]
aflx/L 3]/f sF}rL]n] sf6g\ x' f];\ / k6o\ fOPsf] efunfO{ vf]Ng'xf];\ .
o;/L ags] f cfs[ltdf sltcf]6f ;dldlt /]vfx¿ (Lines of Symmetry) ag,]
5nkmn u/L kQf nufpgx' f;] \ .

kT| os] lrqnfO{ b'O{ a/fa/ efudf af8F g\ ] 86 /]vf (dot line) nfO{ ;dldltsf]
cIf (Axis of Symmetry) jf Pg] f /v] f (Mirror line) klg eGg] u/LG5 .

ul0ft, sIff & 237

pbfx/0f 1
tn /]vf ;dldltsf] cIf / cfwf lrq lbOPsf] 5 . o;nfO{ k"/f ug{'xf;] \ . /v] Lo
;dldltsf] cIfsf] ;ªV\ of klg kQf nufpg'xf;] \ .
;dfwfg
oxfF /v] f ;dldltsf] cIfsf] cfwf/df lrq k"/f ubf{

/]vLo ;dldltsf cIfx¿sf] ;ªV\ of = 2

18.1.2 laGb' ;dldlt (Point Symmetry)
ljm| ofsnfk 3

;a} ljBfyLx{ ¿ pkoS' t ;d"xdf ljeflht eO{ k|Tos] ;dx" n] lbOPsf] lrqnfO{
kf/bzL{ Knfl:6sdf 6];« ugx{' f;] \ .

A

O

B

dflysf] lrqdf l7s ldNg] u/L sG] b| O df kl] G;nsf] 6'Kkfn] lyr]/ 6;]« u/]sf] lrqnfO{
lj:tf/} 3d' fpg'xf];\ .
o;/L 3d' fpFbf,

● slt l8uL| sf] sf0] fdf 3d' fpbF f lrq -cfs[lt_ s]Gbb| l] v a/fa/ b'/Ldf t/
ljk/Lt lbzfdf cfOk'U5 <

● klxns] f] cj:yfdf cfOk'Ubf lrq -cfs[lt_ slt k6s vlK6of] <
● laGb' ;dldltsf] >]0fL slt x'G5 <
● ;fyLx¿;uF 5nkmn u/L lgisif{ sIffdf k|:tt' ug'{xf;] \ .

238 ul0ft, sIff &

lj|mofsnfk 4

cª\uh|] L j0f{dfnfsf sg' s'g cIf/x¿nfO{ laGb' ;dldltsf cfwf/df l7s laraf6
(180° sf] sf]0fdf_ 3'dfpFbf cfsl[ t sG] b| laGb'b]lv a/fa/ b/' Ldf t/ ljk/Lt lbzfdf
xG' 5 <

laGb' ;dldltsf] >]0fL slt xG' 5 < ;fyLx¿;Fu 5nkmn u/L sIffdf k:| tt' ug{'xf;] \ .

sg' } klg cfs[ltnfO{ s'g} lglZrt laGbd' f 180° sf] sf0] fdf 3'dfpFbf s]Gbb| l] v
a/fa/ b/' Ldf t/ ljk/Lt lbzfdf vlK6g] cj:yf cfpg'nfO{ laGb' ;dldlt ePsf]

elgG5 .

pbfx/0f 2 A

lbOPsf] lrqnfO{ laGb' ;dldltsf] cfwf/df sG] b| O df O
3d' fpFbf s:tf] cfs[lt aG5 < B

;dfwfg O
B
oxfF lbOPsf] lrqnfO{ laGb' ;dldltsf] cwf/df -s]Gb| laGb'
O df 180°_ 3'dfpbF f klxn]sf] lrqsf] pN6f] -ljk/Lt A
lbzfdf_ cfsl[ t aG5 .

cEof; 18.1
1. tnsf lrqx¿df 86 /]vfnfO{ ;dldlt cIf dfg]/ k"/f ug{'xf;] \ M
-s_ -v_ -u_

ul0ft, sIff & 239

2. tn lbOPsf lrqdf ;dldltsf] cIf / cfwf lrq lbOPsf] 5 . lrq k/" f ug'{xf;] \ /
/v] Lo ;dldltsf cIfx¿sf] ;ª\Vof klg kQf nufpgx' f];\ M

-s_ -v_ -u_ -3_

3. tnsf k|Tos] laGb' /v] fnfO{ 6]l« ;ª ug'{xf];\ . k|To]sdf /]vLo ;dldltsf] cIf
lvRg'xf;] \ . /v] Lo ;dldltsf] cIf sltcf6] f 5g\, kQf nufpg'xf];\ M

-s_ -v_ -u_

4. tnsf lrqx¿nfO{ 6];« u/L O laGb'df 180° sf] sf0] fdf 3d' fpg'xf;] \ M

-s_ -v_ -u_ -3_
O
OO O

5. tn lbOPsf lrqx¿dWo] s'g s'g lrqx¿df laGb' ;dldlt 5g,\ 56' \ofpg'xf];\ M
-s_ -v_ -u_

-3_ -ª_ -r_

240 ul0ft, sIff &

6. lbOPsf] tf;sf kQLx¿df ePsf] cfsl[ t laGb' ;dldlt xf] jf xfO] g\, xf] eg]
lsg <

kl/ofh] gf sfo{
ljleGg /ªsf sfuh lnO{ To;nfO{ k6o\ fO{ /v] Lo ;dldlt / laGb' ;dld-
lt xg' ] cfs[ltx¿ srF} Ln] sf6/] sIffdf k:| tt' ug'{xf;] \ .

pQ/

(1 – 3) ;Dd lzIfsnfO{ bv] fpgx' f;] \ .
!= s_ @ v_ @ u_ # 3_ % ª_ $ r_ # 5_ $ h_ %

18.2 6;] n] ;] g (Tessellation)

tn lbOPsf lrqx¿ cjnf]sg ugx{' f];\ . tL lrqx¿df s:tf s:tf cfsl[ tx¿ slt
sltcf6] f /x]sf 5g\ < ;uF }sf ;fyL;uF 5nkmn u/L ;r" L tof/ kfg{'xf];\ M

s] tkfO{sF f] 3/df ePsf] gfªn\ f,] sfk{6] , 8f]sf,] OF6fsf jf 9'ª\ufsf] kvfn{ , km'6an
cflbdf o:tf cfs[ltx¿ b]Vge' Psf] 5 < tL cfsl[ tx¿ s;/L /fv]sf 5g\ < ;Fus} f]
;fyL;Fu 5nkmn ug{'xf;] \ .

ul0ft, sIff & 241

lj|mofsnfk 1
;a} ;dx" n] lrqdf bv] fP h:t} u/L /ª\uLg sfuhdf Pp6} gfksf lqe'hfsf/
cfsl[ tx¿ agfO{ sFr} Ln] sf6\gx' f];\ . ca lrqdf bv] fP h:t} u/L tL lqeh' fsf/
6j' m| fx¿ rf6k{ k] /df 6fF:gx' f;] \ . o;/L 6j' |mfx¿ 6fF:bf ;txdf vfnL 7fpF g/xg] u/L
/ gvlK6g] u/L 6f:F g'xf];\ / 6;] ]n];gsf af/]df 5nkmn u/L sIffdf k:| t't ug{'xf;] \ .

Ps jf PseGbf a9L HofldtLo cfs[ltx¿ gvK6fOsg / vfnL 7fpF g/fvLsg
;dtn ;tx 9fSg] jf 5f]Kg] k|lj|mofnfO{ 6];n] ];g (Tessellation) elgG5 .

pbfx/0f 1
lbOPsf yf]Knfx¿ hf]8/] lqeh' fsf/ 6;] ]n;] g agfpgx' f];\ / /ª egx{' f];\ .

;dfwfg
dflysf yf]Knfx¿ hf8] /] lgDgfg';f/sf] lqeh' fsf/ 6];]n;] g agfpg ;lsG5 .

242 ul0ft, sIff &

cEof; 18.2
1. lbOPsf lrqx¿ lqe'hfsf/ 6;] n] ;] g xf] jf xf]Og / lsg, kQf nufpg'xf;] \ .
-s_ -v_ -u_
2. tn bv] fOPsf] h:t} u/L lqe'hfsf/ 6;] n] ;] g agfO{ k"/f ug'x{ f;] \ .

3. lbOPsf yfK] nfx¿ hf]8]/ lqeh' fsf/ 6;] n] ;] g agfpgx' f];\ / /ª eg'x{ f;] \ .

4. ;dsf0] f lqeh' / ;dafx' lqeh' af6 aGg] Ps Pscf6] f 6];]n;] g agfpg'xf];\ .

kl/of]hgf sfo{

cfkm\gf] ljBfno, 3/sf] leQf, 3/df ko| fu] u/Lg] afy?dsf 6fon, sfk]{6,
km6' an / elnan cflbdf agfOPsf lrqx¿ cjnf]sg u/L lqe'hfsf/
6];]n;] gx¿sf] lrq cfkm\gf] sfkLdf agfpg'xf;] \ / /ª;dt] e//] sIffdf
k:| t't ugx{' f;] \ .

pQ/

lzIfsnfO{ b]vfpg'xf];\ .

ul0ft, sIff & 243