Com. Mathematics (VII)

b'O{cf]6f sf0] fsf] ofu] kmn b'O{ ;dsf]0f jf 180° xG' 5 eg] tL bO' c{ f6] f
P
sf0] fnfO{ Pscsf{sf kl/k/" s sf0] f elgG5 .

;uF s} f] lrqdf bO' { sf0] f ∠PQR = 70° / ∠ABC = 110° QR

5 . oL b'O{ sf0] fsf] hf8] klg 180° xG' 5 . A
B
ctM ∠PQR sf] kl/k"/s sf0] f ∠ABC xf] . C

pbfx/0f 1

lbOPsf] lrqdf AB / CD laGb' O df sfl6Psf 5g\ eg] M

CB

O
AD

-s_ ∠AOD / ∠AOC sf zLiff{ledv' sf]0fx¿ n]Vgx' f];\ .
-v_ ∠AOD sf] cf;Gg sf]0fx¿ s'g s'g x'g,\ n]Vgx' f];\ .
-u_ ∠BOD / ∠BOC sf] of]ukmn slt l8u|L xG' 5, nV] g'xf];\ .
-3_ ∠AOC ;Fu a/fa/ xg' ] sf]0f sg' xf], n]Vgx' f;] \ .
-ª_ ∠BOC sf] kl/k/" s sf]0f n]Vg'xf;] \ .
;dfwfg
oxfF,
-s_ ∠AOD sf] zLiffl{ ed'v sf]0f ∠BOC / ∠AOC sf] zLiff{ledv' sf0] f ∠BOD

xg' \ .
-v_ ∠AOD sf] cf;Gg sf0] fx¿ j|mdzM ∠AOC / ∠BOD xg' \ .
-u_ ∠BOD / ∠BOC sf] of]ukmn 180° x'G5 .
-3_ ∠AOC ;uF a/fa/ x'g] sf0] f ∠BOD xf] .
-ª_ ∠BOC sf] kl/k/" s sf]0fx¿ ∠AOC / ∠BOD bj' } x'g\ .

ul0ft, sIff & 195

pbfx/0f 2

lbOPsf] lrqdf PQ, QR / QS laGb' Q af6 lvlrPsf 5g\ h;df ∠PQR = 90° 5 eg],

-s_ ∠PQS sf] cf;Gg sf0] f n]Vg'xf];\ .

-v_ ∠SQR sf] ;dk/" s sf]0f nV] gx' f];\ . P
;dfwfg

oxf F -s_ ∠PQS sf] cf;Gg sf]0f ∠SQR xf] . S
-v_ ∠SQR sf] ;dk"/s sf]0f ∠PQS xf] . R

pbfx/0f 3 Q

33° sf0] fsf] ;dk"/s / kl/k"/s sf0] fx¿ kQf nufpg'xf];\ M
;dfwfg
oxfF lbOPsf] sf0] f = 33°
cyjf 33° sf]0fsf] ;dk/" s = (90° – 33°) = 57°

cyjf 33° sf]0fsf] kl/k/" s = (180° – 33°) = 147°

ctM 33° sf] ;dk"/s sf]0f 57° / kl/k"/s sf0] f 147° xf] .

pbfx/0f 4

lbOPsf] lrqaf6 x, y / a sf] dfgx¿ lgsfNgx' f];\ M A D
(3a)o
yo O (2a + 40)o
xo

;dfwfg CB
oxfF,

-s_ ∠AOC = ∠BOD ⸪ zLiff{ledv' sf]0fx¿
cyjf 3a = 2a + 40°

cyjf a = 40°

-v_ ∠AOD + ∠AOC = 180° ⸪ l;wf /v] fdf ags] f sf0] f

196 ul0ft, sIff &

cyjf y + 3a = 180° ⸪ a = 40°
cyjf y + 3 × 40 = 180° ⸪ zLiffl{ ed'v sf]0fx¿
cyjf y = 180° – 120°
cyjf y = 60°
-u_ ∠BOC = ∠AOD
cyjf x = y
cyjf x = 60°
ctM a = 40o, y = 60o / x = 60o

cEof; 13.3

1. lbOPsf] lrqdf ∠AXC ;uF tnsf ;DaGw /xs] f sf0] fx¿ C B
nV] gx' f;] \ M X

-s_ b'Oc{ f6] f cf;Gg sf0] f A D
-v_ b'Oc{ f]6f kl/k"/s sf]0f
-u_ Pp6f zLiff{ledv' sf]0f A B
C
2. lbOPsf] lrqdf ∠AOB = 90° 5 .

-s_ ∠AOC sf] cf;Gg sf0] f n]Vgx' f];\ .
-v_ ∠BOC sf] ;dk/' s sf]0f n]Vg'xf;] \ . O
3. tn lbOPsf sf]0fsf] ;dk"/s / kl/k/" s sf]0fx¿ n]Vg'xf;] \ M

-s_ 15° -v_ 45° -u_ 78° -3_ 87°

4. tn lbOPsf lrqaf6 x, y / a sf] dfg kQf nufpgx' f;] \ M

-s_ A -v_ R

C 140° (x + 15)°
B
PO Q

(2a)o
ao

O

ul0ft, sIff & 197

-u_ P S -3_ A xo D

88o (3x)o 80° O yo
xo T ao

RQ

-ª_ CB

40° 40°
y°
30° x°

kl/ofh] gf sfo{

bO' c{ f]6f l;wf /]vfx¿ Pscfk;df kl| tR5b] g x'Fbf aGg] ;Defljt hf8] f
sf0] fx¿sf af/]df 5nkmn u/L sIffdf k:| tt' ug{'xf];\ .

pQ/ -v_ ∠AXD / ∠CXB -u_ ∠BXD

1. -s_ ∠BXC / ∠AXD -v_ ∠AOC
2. -s_ ∠BOC
3. -s_ 75°, 165° -v_ 45° , 135°
-u_ 12° , 102°
4. -s_ 30° -3_ 3° , 93°
-3_ 100° , 80°, 100°
-v_ 25° -u_ 23°

-ª_ 50°, 20°

198 ul0ft, sIff &

13.4 s f0] fx¿sf] k|of]ufTds k/LIf0f (Experimental Verification of

Angles)
k/LIf0f 1

b'O{cf]6f l;wf/]vfx¿ Pscfk;df sfl6Fbf aGg] zLiffl{ ed'v sf]0fx¿ a/fa/ x'G5g\ .

oxf,F RQ SQ

OO

PS PR

lrq 1 lrq 2

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

lrq ∠ROP ∠QOS ∠ROQ ∠POS kl/0ffd

1
2

lgisif{M bO' c{ f6] f l;wf/v] fx¿ Pscfk;df sfl6bF f aGg] zLiffl{ ed'v sf0] fx¿
a/fa/ xG' 5g\ .

k/LIf0f 2

Pp6f l;wf/v] fn] csf]{ l;wf/v] f;uF 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 sg' } laGb' Q af6 QP /]vfv08 lvrL km/s
km/s b'O{cf6] f lrq lvRgx' f;] \ . ca rfFbsf] ;xfotfn] sf]0fx¿ j|mdzM ∠PQA /
∠PQB nfO{ gfKgx' f;] \ / tnsf] tflnsfdf eg'x{ f;] \ .

ul0ft, sIff & 199

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
sf0] fx¿sf] of]ukmn 180° xG' 5 .

k/LIf0f 3
s'g} laGb'sf] jl/kl/ Ps kl/j|md0fdf ag]sf sf0] fx¿sf] of]ukmn 360° xG' 5 .

AA

BO B OC

C lrq 1 lrq 2

dflysf] h:t} cfˆgf] sfkLdf b'Oc{ f6] f lrq lvRgx' f;] \ / ∠AOB, ∠BOC jx[ t\ sf0] f
∠AOC nfO{ gfKgx' f];\ / tnsf] tflnsfdf egx{' f];\ .

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

∠AOC
1

2

lgisif{ M s'g} laGb'sf] jl/kl/ Ps kl/jm| d0fdf ag]sf sf0] fx¿sf] ofu] kmn
360° x'G5 .

200 ul0ft, sIff &

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 /]vfdf ag]sf cf;Gg
cyjf 3a = 90° sf]0fx¿sf] of]ukmn 180° x'G5 .

cyjf a = 30°

pbfx/0f 2

lbOPsf] lrqaf6 x sf] dfg kQf nufpgx' f];\ M

R
Q

40°
S x°

80° O (3x)°

;dfwfg P

oxfF ∠POQ + ∠ROQ + ∠ROS + ∠SOP = 360° ⸪ sg' } laGb'df jl/kl/
ag]sf sf0] fx¿sf]
cyjf 3x + 40° + x + 80° = 360°
ofu] kmn 360° xG' 5 .
cyjf 4x + 120° = 360°

cyjf 4x = 360° – 120°

cyjf 4x = 240

cyjf x = 2440
cyjf x = 60°

ul0ft, sIff & 201

pbfx/0f 3

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

;dfwfg Q

oxf F ∠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 nufpgx' f];\ . 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

202 ul0ft, sIff &

2. tn lbOPsf tYox¿sf] k/LIf0fåf/f k|dfl0ft ugx'{ f;] \ .
-s_ b'O{cf6] f l;wf/]vfx¿ Pscfk;df sf6b\ f aGg] zLiff{led'v sf]0fx¿
a/fa/ xG' 5g\ .
-v_ l;wf /]vfsf] sg' } laGb'df Ps}lt/ ags] f cf;Gg sf]0fx¿sf] of]ukmn 180°
x'G5 .
-u_ sg' } laGb'sf] jl/kl/ Ps kl/j|md0fdf ags] f sf0] fx¿sf] of]ukmn 360°
x'G5 .

pQ/

1. -s_ 27° -v_ y = 50°, x = 110°, b = 70°
-u_ x = 20° -3_ 40°
2. ;a} k|Zgsf] ;dfwfg sIffdf 5nkmn ug'{xf];\ .

ul0ft, sIff & 203

kf7 14 ;dtnLo cfs[ltx¿

(Plane Figures)

14.0 kg' /jnfs] g (Review)

tn lbOPsf lqeh' x¿ -;dafx', ljifdafx', Go"gsf0] fL, ;dsf0] fL, clwssf0] fL_ s'g
ks| f/sf x'g\, sIffdf 5nkmn ug{'xf];\ .

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

-3_ -ª_ -r_

60° 90° 105°
50° 70°

14.1 lqeh' sf] /rgf (Construction of Triangle)

14.1.1 b'Oc{ f]6f eh' fx¿ / ltgLx¿ larsf] sf0] f lbOPdf lqe'hsf] /rgf
lj|mofsnfk 1

PQ = 5.6 cm, QR = 4.5 / ∠PQR = 60° ePsf] Pp6f lqeh' PQR sf] /rgf

ugx{' f;] \ M R

;jk{ y| d v];|f lrq lvRg'xf;] \ . 4.5 cm

ljlw

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] sf0] f

lvRg'xf];\ .

204 ul0ft, sIff &

3. laGb' Q af6 QR = 4.5 cm sf] gfk 4.5 cm R
lnP/ sf6g\ 'xf];\ . Q 600

4.= laGb' R / P nfO{ hf]8g\ x' f];\ . 5.6 cm

ctM cfjZos lqeh' PQR xf] .

P

14.1.2 sg' } Pp6f eh' f / To;df ags] f bO' c{ f6] f sf]0fx¿ lbOPdf lqeh' sf] /rgf

lj|mofsnfk 2

AB = 5.2 cm, ∠A = 75° / ∠B = 60° ePsf] ∆ABC sf] /rgf ugx'{ f];\ .

;j{k|yd v;] |f lrq lvRg'xf;] \ . C
ljlw

1. AB = 5.2 cm sf] Pp6f /]vfv08 lvRg'xf;] \ .

2. laGb' A df sDkf;sf ;xfotfn] 75° sf] sf]0f lvRgx' f;] \ .

3. laGb' B df sDkf;sf ;xfotfn] 60° sf] sf]0f lvRg'xf];\ . A 75° 60° B
5.2 cm
4. o;/L 75° / 60° agfPsf /v] fx¿ sfl6Psf] laGb'sf] gfd C

lbg'xf;] \ . C
ca cfjZos lqeh' ∆ABC xf] .

A 75° 5.2 cm 60° B

ul0ft, sIff & 205

14.1.3 tLgcf6] } e'hf lbOPdf lqeh' sf] /rgf

ljm| ofsnfk 3

AB = 4.5 cm, BC = 5 cm / CA = 6.5 cm ePsf] lqeh' ABC sf] /rgf ugx'{ f;] \ M

;jk{ y| d v];|f lrq lvRg'xf;] \ . C
ljlw
6.5 cm
1. AB = 4.5 cm sf] Pp6f /v] f v08 lvRg'xf];\ . 5 cm

2. laGb' A af6 6.5 cm gfksf] cwJ{ of; / laGb' B af6 5 cm A 4.5 cm B
gfksf] cw{Jof; lng'xf;] \ / Pp6} laGb'df sfl6g] u/L rfk

sf6g\ 'xf;] \ .

3. tL bO' c{ f6] f rfk sfl6Psf] laGb'sf] gfd C lbgx' f];\ . C

4. /]vf A / C tyf B / C hf8] \gx' f];\ .

ca cfjZos lqe'h ABC sf] /rgf xf] . 6.5 cm
5 cm

A B
4.5 cm

14.1.4 sg' } Pp6f eh' f, To;df cfwfl/t Pp6f sf]0f / To; e'hfsf] ;Dd'v sf0] f
lbOPdf lqeh' sf] /rgf

ljm| ofsnfk 4

XY = 5 cm, ∠ZXY = 45o / ∠XZY = 75o ePsf] ∆XYZ sf] /rgf ug{x' f];\ M

tn lbOPsf r/0fx¿ ckgfO{ AB = 4.5 cm, BC = 5 cm / CA = 6.5 cm ePsf] lqeh'

ABC sf] /rgf ugx{' f];\ M Z

;jk{ y| d v];f| lrq lvRg'xf;] \ . 75o

X 45o Y
5 cm

206 ul0ft, sIff &

ljlw

1. XY = 5 cm sf] Pp6f /]vf v08 lvRgx' f;] \ .

2. laGb' x df sDkf;sf] ;xfotfn] 45o sf] sf0] f lvRgx' f;] \ .

3. laGb' Y df [180o – (75o+45o) = 180o – 120o] 60o sf] sf]0f lvRg'xf;] \ .

4. o;/L laGb' X / Y df 45o / 60o sf] sf0] fx¿ agfPsf /v] fx¿ sfl6Psf]
laGb'sf] gfd Z lbgx' f];\ .

ca cfjZos lqeh' XYZ tof/ eof] .

Z

45o 60o Y

X 5 cm

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'x{ f];\ M

-s_ DE = 4.5 cm, EF = 4 cm / DF = 5 cm
-v_ EF = 6.6 cm, DF = 6 cm / DE = 7 cm
-u_ DE = EF = 5.5 cm, DF = 5.2 cm

ul0ft, sIff & 207

4. tn lbOPsf cj:yfdf ∆LMN sf] /rgf ugx'{ f;] \ M
-s_ LM = 6 cm, ∠NLM = 60O / ∠LNM = 90O
-v_ MN = 5.5 cm, ∠LMN = 45O / ∠MLN = 60O
-u_ LN = 7 cm, ∠MLN = 60O / ∠LMN = 90O

5. pko'St ;ª\Vofdf ;dx" df 5nkmn u/L tn lbOPcg;' f/sf efusf] gfk lnP/
lqe'hsf] /rgf ug{sf nflu k|Zgx¿ lgdf0{ f ugx{' f];\ / /rgf u/L sIffdf k|:tt'
ug'x{ f;] \ .
-s_ b'Oc{ f]6f eh' f / ltgLx¿larsf] sf0] fsf] gfk
-v_ tLgcf6] f eh' fsf] nDafOsf] gfk
-u_ Pp6f eh' f / To; eh' fdf ags] f b'Oc{ f6] f sf0] fsf] gfk

pQ/

lzIfsnfO{ b]vfpg'xf;] \ .

14.2 ;dfgfGt/ rte'{ 'h, cfot / jus{ f u0' fx¿sf] klxrfg / k/LIf0f
(Identification and Verification of the Properties of

Parallelogram, Rectangle and Square)

lj|mofsnfk 1

Pp6f lstfa, Sof/d] af8] { / sk8fsf] 58\s] sf6s] f] 6'j|mf lngx' f;] \ / tL 7f;] j:ts' f
;txsf cfsf/x¿ s] s:tf 5g\, sIffdf 5nkmn ug{'xf;] \ .

lstfa Sof/d] af8] { sk8f

tL ;txsf lsgf/f / s'gfdf ags] f sf0] fx¿ gfKgx' f];\ . lsgf/f / sf]0fx¿sf
cfwf/df rte'{ 'hsf] u'0faf/] sIffdf 5nkmn ugx{' f];\ .

208 ul0ft, sIff &

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

Properties of Parallelogram)

ljm| ofsnfk 2

lbOPsf] ;dfgfGt/ rt'{e'h PQRS sf ;a} eh' fx¿, sf0] fx¿ ljs0fs{ f efux¿
gfKgx' f;] \ . ca e'hfx¿sf] ;DaGw, sf]0fx¿sf] ;DaGw / ljs0f{sf efux¿sf] ;DaGw
s:tf] bV] ge' of] sIffdf ;fyLx¿;uF 5nkmn u/L ;dfgfGt/ rt{e' h' sf u'0fx¿ kQf
nufpgx' f;] \ .

SR

PQ

k/LIf0f 1

;dfgfGt/ rte' h'{ sf ;Ddv' sf]0fx¿ a/fa/ x'G5g\ egL k/LIf0f ug'{xf];\ .

S RPQ

P lrq 1 Q S R

lrq 2

lbOPsf bO' c{ f6] f ;dfgfGt/ rte' {h' PQRS sf ;a} sf0] fx¿sf gfk lnP/ tnsf]
tflnsfdf eg{'xf];\ .

lrq ∠QPS ∠PQR ∠QRS ∠RSP kl/0ffd

1
2

lgisif{ M ;dfgfGt/ rte'{ h' sf] ;Dd'v sf0] fx¿ a/fa/ xG' 5g\ .

ul0ft, sIff & 209

k/LIf0f 2

;dfgfGt/ rte' h'{ sf ;Dd'v eh' fx¿ a/fa/ xG' 5g\ egL k/LIf0f ugx{' f;] \ .

S R PQ

P Q SR

lrq 1 lrq 2

lbOPsf b'O{cf6] f ;dfgfGt/ rt'eh{' PQRS sf ;a} eh' fx¿sf] gfk lnP/ tnsf]

tflnsfdf eg'{xf];\ .

lrq PQ QR RS SP kl/0ffd

1
2

lgisif{ M ;dfgfGt/ rt'{eh' sf] ;Ddv' eh' fx¿ a/fa/ x'G5g\ .
k/LIf0f 3

;dfgfGt/ rte{' h' sf] ljs0f{x¿ ;dlåefhg xG' 5g\ . C

A DD
OO

B lrq 1 C AB

lrq 2

lbOPsf bO' {cf]6f ;dfgfGt/ rte'{ h' df ljs0f{x¿ AC / BD laGb' O df sfl6Psf
5g\ . ca ljs0f{sf efux¿sf] nDafO gfkL tn lbOPsf] tflnsfdf eg{x' f;] \ .

lrq AO OC BO OD kl/0ffd

1
2

lgisif{ M ;dfgfGt/ rte{' h' sf] ljs0f{x¿ ;dlåefhg xG' 5g\ .

210 ul0ft, sIff &

;dfgfGt/ rt'eh'{ sf u0' fx¿
-s_ ;dfgfGt/ rt'{eh' sf ;Ddv' sf0] fx¿ a/fa/ x'G5 .
-v_ ;dfgfGt/ rte{' 'hsf ;Dd'v eh' fx¿ a/fa/ xG' 5 .
-u_ ;dfgfGt/ rt{e' h' sf ljs{0fx¿ k/:k/ ;dlåefhg x'G5g\ .

14.2.2 cfotsf u'0fx¿sf] klxrfg
(Identification of the Properties of Rectangle)

lj|mofsnfk 3 S R
P Q
Pp6f sfkLsf] kfgf lngx' f;] \ / o;sf ;Dd'v lsgf/fx¿ /
s'gfdf ePsf sf0] fx¿ / ljs0fx{ ¿sf] nDafO gfKgx' f;] \ .
o;/L gfKbf s] u0' f kQf nfU5, sIffdf ;fyLx¿;uF 5nkmn
ugx'{ f];\ .

k/LIf0f 1

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

R

P Q

lrq 1 lrq 2 S

lbOPsf b'Oc{ f6] f cfot PQRS sf kT| os] sf]0fx¿ gfkL tn lbPsf] tflnsfdf
egx{' f];\ .

lrq ∠QPS ∠PQR ∠QRS ∠RSP kl/0ffd

1
2

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

ul0ft, sIff & 211

k/LIf0f 2

cfotsf ;Ddv' eh' fx¿ a/fa/ xG' 5g\ . S

SR

P
R

P lrq 1 Q lrq 2

Q

lbOPsf b'Oc{ f6] f cfot PQRS sf ;a} e'hfx¿ gfkL tn lbOPsf] tflnsfdf
eg'x{ f];\ / ltgLx¿larsf] ;DaGw kQf nufpg'xf;] \ .

lrq PQ QR RS SP kl/0ffd

1
2

lgisif{ M cfotsf ;Ddv' e'hfx¿ a/fa/ xG' 5g\ .

k/LIf0f 3 C
cfotsf ljs0{ fx¿ a/fa/ x'G5g\ .
D B
AD
lrq 2 A
BC

lrq 1

lbOPsf bO' {cf]6f cfot ABCD sf ljs0f{x¿ AC / BD sf gfk lnO{ tn lbOPsf]
tflnsfdf eg'x{ f;] \ .

lrq AC BD kl/0ffd

!

@

lgisif{ M cfotsf ljs{0fx¿ a/fa/ x'G5g\ .

212 ul0ft, sIff &

cfotsf u0' fx¿
-s_ cfotsf ;a} sf0] fx¿ 90° sf xG' 5g\ .
-v_ cfotsf ;Dd'v eh' fx¿ a/fa/ xG' 5g\ .
-u_ cfotsf ljs0fx{ ¿sf] nDafO a/fa/ xG' 5g\ .

14.2.3 jus{ f u0' fx¿sf] klxrfg
(Identification of the Properties of Square)

lj|mofsnfk 4

Pp6f ldgL r;] sf] af8] { jf juf{sf/ ;tx ePsf] 7f;] j:t' lngx' f;] \ . pSt
r];sf] af]8{ jf 7f;] j:tx' f] jufs{ f/ ;txnfO{ sfkLdfly /fvL jl/kl/ 3/] f
nufpgx' f];\ / ljs0fx{ ¿ hf]8g\ x' f];\ . o;/L ags] f] rt{e'hsf ;a} eh' fx¿, sf]0fx¿,
ljs0fs{ f] nDafO, ljs0f{larsf] sf0] f, ljs0f{sf efux¿ / zLif{sf]0fsf ljeflht
sf]0fx¿ gfKg'xf];\ . o;/L gfKbf s] kl/0ffd kQf nfU5 sIffdf ;fyLx¿;Fu 5nkmn
ug'x{ f;] \ .

k/LIf0f 1

jus{ f ;a} sf]0f / e'hf a/fa/ xG' 5g\ . W

XY X

WZ Z

lrq 1 Y

lrq 2

lbOPsf bO' {cf6] f 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 & 213

k/LIf0f 2 R
ju{sf ljs0f{x¿sf] nDafO a/fa/ x'G5g\ . Q

PS
S

Q R P

lrq 1 lrq 2

lbOPsf b'Oc{ f6] f ju{ PQRS sf ljs0fx{ ¿ PR / QS 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 nDafws{ xG' 5g\ . R

PS

O

SO Q

Q R lrq 2 P

lrq 1

lbOPsf bO' {cf6] f ju{ PQRS sf ljs0f{sf efux¿ / ljs0f{x¿larsf sf0] fx¿sf] gfk
lnO{ tnsf] tflnsfdf eg{x' f;] \ .

214 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 ljs0f{n] zLif{sf0] fnfO{ cfwf ub{5 . R

PS

O S Q
O

Q R

lrq 1 lrq 2 P

lbOPsf b'Oc{ f]6f ju{ PQRS sf zLif{sf0] fx¿ / ljs0fx{ ¿n] ljefhg u/]sf zLifs{ f0] fsf
efux¿sf] gfk lnO{ tnsf] tflnsfdf eg'{xf];\ M

lrq 1 ;xfos sf]0fx¿sf] gfk kl/0ffd
zLifs{ f0] fsf] gfk

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

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

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

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

ul0ft, sIff & 215

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

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

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

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

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

lgisif{ M ju{sf kT| os] ljs0fn{ ] sf0] fnfO{ cfwf ub{5 .

ju{sf u'0fx¿
-s_ ju{sf ;a} sf0] fx¿ / eh' fx¿ cfk;df a/fa/ x'G5g\ .
-v_ jus{ f ljs0fx{ ¿sf] nDafO cfk;df a/fa/ xG' 5g\ .
-u_ jus{ f ljs0f{x¿ cfk;df nDafw{ s xG' 5g\ .
-3_ ju{sf ljs0f{x¿n] zLif{ sf]0fx¿nfO{ cfwf ub5{ .

cEof; 14.2

tn lbOPsf egfO l7s 5g\ jf 5g} g\ 56' o\ fpg'xf;] \ .
-s_ ;a} rte' h{' sf ;Dd'v ehfx¿ a/fa/ x'G5g\ .
-v_ ;dfgfGt/ rt'e'h{ sf ;Ddv' sf0] fx¿ a/fa/ xG' 5 .
-u_ jus{ f ;Dd'v e'hfx¿ dfq a/fa/ x'G5g\ .
-3_ cfotsf ljs0fx{ ¿ cfk;df nDafws{ xG' 5g\ .
-ª_ cfotsf ;a} u0' f ;dfgfGt/ rte' 'h{ df klg x'G5 .
-r_ jus{ f ljs0f{x¿ a/fa/ xG' 5g\ .
-5_ cfotsf ;Dd'v sf0] fx¿ dfq a/fa/ xG' 5g\ .

216 ul0ft, sIff &

kl/ofh] gf sfo{

cfˆgf] 3/ / ljBfno jl/kl/ /x]sf cfotfsf/, jufs{ f/ / ;dfgfGt/
rt'e'{h cfsf/sf ;txx¿ ePsf j:tx' ¿ vfH] gx' f];\ . pSt j:ts' f
;txx¿ sfkLdf 6];« u/L cfot, ju{ / ;dfgfGt/ rte' {'hsf u'0fx¿
k/LIf0f ugx'{ f];\ / sIffdf k|:t't ugx'{ f;] \ .

pQ/

lzIfsnfO{ b]vfpgx' f];\ .

14.3 kfOyfuf]/; ;fWo (Pythagoras Theorem)

kfOyfuf/] ; ;fWosf] k/LIf0f

Pp6f ;dsf]0fL lqe'h lvRgx' f];\ h;df ∠Q = P h
90° 5 . pSt ;dsf]0f lqeh' sf] s0f{ (h), nDa P
(P) / cfwf/ (b) 5 .

;dsf0] f lqeh' sf] s0f{ (h), nDa (P) / cfwf/
s;/L kQf nufpg ;lsG5 <

pSt ;dsf]0fL lqeh' sf] e'hfx¿ gfKgx' f];\ / Q b R
kT| o]s e'hfdf Ps Pscf]6f ju{ agfpg'xf;] \ . A

ca ju{ A, B / C sf] If]qkmn lgsfNgx'\ f];\ . P R
sIffdf ;fyLx¿;Fu 5nkmn ug{'xf;] \ . s] h df B
ag]sf] jus{ f] Ifq] kmn = P df ag]sf] jus{ f] Ifq] kmn
+ b df ags] f] Ifq] kmn x'G5, 5nkmn ug{'xf;] \ .

oxfF s0fd{ f ag]sf jus{ f] If]qkmn cGo b'O{ eh' fdf Q
ags] f jus{ f] ofu] kmn;Fu a/fa/ eof] . C

ctM -s0f_{ 2 = -nDa_2 + -cfwf/_2

h2 = p2 + b2

hxfF h s0f{ (hypotenuse), p nDa
(perpendicular) / b cfwf/ (base) 5g\ .

ul0ft, sIff & 217

;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| o]s lsgf/fsf] nDafO
gfKg'xf;] \ . ;ae} Gbf nfdf] lsgf/f 5'6o\ fpg'xf;] \ . ;aeGbf nfdf]
lsgf/fsf] gfksf] ju{;uF cGo b'O{ lsgf/fsf] gfksf] jus{ f]
ofu] kmn a/fa/ xG' 5 jf xb' }g sIffdf 5nkmn ugx'{ f];\ / lgisif{
lgsfNg'xf];\ .

pbfx/0f 1

lbOPsf] lqe'h ;dsf0] fL xf] jf xfO] g kQf nufpg'xf;] \ . 12 cm 13 cm
;dfwfg 5 cm
dfgf}F 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 b'O{ e'hfsf] jus{ f] ofu] kmn a/fa/ eof] . ctM pSt lqeh'
;dsf0] fL lqeh' xf] .

218 ul0ft, sIff &

pbfx/0f 2

lbOPsf] ;dsf]0fL lqe'hdf yfxf gePsf] e'hf kQf nufpgx' f];\ M

;dfwfg

oxfF ∠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] eh' f 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} lqe'hsf] nfdf] eh' f s0f{ xG' 5 .
-v_ ;dsf0] fL lqehdf dfq kfOyfuf/] ; ;fWo dfGo x'G5 .
-u_ ;dsf]0fL lqe'hdf ;dsf0] f agfpg] Pp6f e'hf s0f{ x'g'k5{ .
-3_ ;dsf]0fsf] ;Ddv' e'hf lqe'hsf] s0f{ x'G5 .

2. tn lbOPsf gfk ePsf sg' sg' lqe'h ;dsf]0fL lqeh' 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 & 219

3. tn lbOPsf ;dsf0] fL lqe'hdf yfxf gePsf] e'hfsf] 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/ofh] gf sfo{

cfˆgf] 3/ / ljBfno jl/kl/ /x]sf lqeh' cfsf/sf ;tx ePsf j:tx' ¿ vfh] L
tL ;txsf lsgf/fsf] nDafOsf gfk lng'xf];\ / 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_ xf]Og -3_ xf]
2. -s_ xf]Og
3. -s_ 4 cm -v_ 8 cm -u_ 13 cm -3_ 15 cm -ª_ 13 cm

220 ul0ft, sIff &

kf7 15 cg'¿k cfsl[ tx¿

(Congruent Figures)

15.0 k'g/jnf]sg (Review)

tn lbOPsf cfsl[ tx¿dWo] sg' sg' p:t} cfsf/sf / a/fa/ gfksf 5g,\ 56' o\ fpgx' f];\ M

15.1 cg'¿k cfs[ltx¿ (Congruent Figures)

ljm| ofsnfk 1

sIffsf ;Dk0" f{ ljBfyL{nfO{ rf/ ;dx" df ljefhg ug{ kT| o]s ljBfyLn{ ] cfcfˆgf]
Hofldlt afs;df ePsf] ;]6:\ SjfP/ lnP/ cfˆgf] ;dx" sf] ;fyLsf] ;6] :SjfP/ dfly
/fVg'xf];\ -vK6o\ fpg'xf;] \_ / tn' gf u/L tnsf k|Zgx¿sf] pQ/ vf]Hg'xf];\ M

vK6\ofPsf ;6] :Sjfo/

ul0ft, sIff & 221

-s_ ;6] :Sjfo/x¿ s:tf cfsf/sf 5g\ <

-v_ vK6\ofPsf ;]6 :Sjfo/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|bzg{ ugx'{ 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'jm| f lng'xf];\ . lrqdf
b]vfP h:t} u/L l7s lardf k6o\ fpgx' f];\ . k6o\ fPsf]
sfuhnfO{ vf]n/] k6l\ 6Psf] 7fpdF f s}FrLn] sf6\gx' f;] \ .
bj' } 6'j|mfnfO{ vK6\ofpgx' f;] \ / t'ngf u/L ;d"xdf tnsf
k|Zgx¿sf] pQ/ vf]hL u/L sIffdf k:| tt' ug{'xf;] \ .
-s_ bj' } cfsl[ tx¿ p:t} cfsf/sf 5g\ ls 5}gg\ <
-v_ bj' } 6j' |mfsf gfkx¿ a/fa/ 5g\ ls 5g} g\ <
-u_ cfsf/ p:t} / a/fa/ gfk ePsf cfs[ltnfO{ s:tf cfs[lt elgG5 <

ljm| ofsnfk 3

Pp6f aG] rdf a;s] f ;fyLx¿sf] Pp6f ;dx" x'g] u/L ;dx" df ljefhg eO{ tn
lbOPsf kT| os] hf]8L lrqx¿nfO{ 6«]l;ª k]k/sf] ;xfotfn] sfkLdf agfpg'xf;] \ .

AP M PA D

B CQ RN OB C

cfkmn" ] agfOPsf tL kT| os] hf8] L lrqsf] aflx/L 3]/f s}rF Lsf] ;xotfn]
sf6g\ 'xf];\ . To;kl5 Pp6f lrqdfly csf]{ lrq vK6\ofO{ tn' gf ugx'{ f];\ / tnsf kZ| gsf]
pQ/ vfH] g'xf];\

222 ul0ft, sIff &

-s_ klxnf] hf8] L lrqx¿ -lqe'hx¿_ p:t} cfsf/sf 5g\ ls 5g} g\ <
-v_ klxnf] hf]8L lrqx¿ -lqeh' x¿_ sf gfk a/fa/ 5g\ ls 5g} g\ <
-u_ To:t} bf;] |f] hf8] L lrqx¿ -rt'e'h{ x¿_ df s] s] s'/fx¿df ;dfgtf 5 <
;fyL;Fu 5nkmn ug'x{ f;] \ / lgisif{ sIffdf k|:t't ug{'xf];\ .

lj|mofsnfk 4

kT| os] ljBfyLn{ ] Ps Pscf]6f ;]6 :Sjfo/ lnO{ cfcfˆgf] sfkLdf -;6] :\ Sjfo/_
/fv]/ To;sf] aflx/L 3]/f 6«]; u/L bO' { b'Oc{ f6] f lqe'hx¿ agfpgx' f];\ . kT| o]s lqeh' sf]
gfdfª\sg j|mdzM ABC / XYZ ug{'xf;] \ .

AX

B CY Z

k|Tos] lrqsf] aflx/L 3/] f sf6]/ lqeh' ABC nfO{ lqe'h XYZ dfly /fvL bfHF g'xf];\ /
tnsf] tflnsf eg{'xf];\ M

lqeh' 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 eh' f ===================5 .
eh' f YZ dfly e'hf=====================5 .
eh' f ZX dfly eh' f ==================5 .
XY = .........., YZ = ............. / ZX = ................. 5 .

ul0ft, sIff & 223

lqeh' ABC / lqe'h XYZ nfO{ s:tf lqe'hx¿ eGg ;lsG5 < ;fyL;Fu 5nkmn
ugx{' f;] \ / lgisif{ sIffdf k:| t't ug'x{ f;] \ M

p:t} cfsf/ / a/fa/ gfk ePsf cfsl[ tx¿nfO{ cg'¿k cfsl[ t
(Congruent 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 cfs[ltx¿ -c_ / -cf_ cg'¿k 5g\ lsgls ltgLx¿sf cfsf/ p:t} 5g\ / e'hfsf
gfkx¿ klg a/fa/ 5g\ .

pbfx/0f 2
tnsf cfs[ltx¿ cjnf]sg ubf{ sg' s'g cfsl[ tx¿ cg'¿k 5g\, lsg <
-c_ -cf_ -O_

;dfwfg

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

224 ul0ft, sIff &

cEof; 15

1. tnsf sg' s'g cfs[ltx¿ cg'¿k 5g,\ lsg < 2 cm

3 cm

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

3 cm 2 cm

-v_ -c_ -cf_

6 cm
6 cm
6 cm
6 cm

6 cm 6 cm

-u_ -c_ 4cm -cf_ 5cm
5cm
2cm 2cm 3cm 3cm

4cm

2. tnsf sg' sg' cfs[ltx¿ cg¿' k 5g\ < gfk lnP/ kQf nufpg'xf];\ .

s_ -c_ -cf_

-v_ -c_ -cf_

-u_ -c_ -cf_

ul0ft, sIff & 225

-3_ -c_ -cf_

-ª_ -c_ -cf_

-r_ -c_ -cf_

-5_ -c_ -cf_

3. kfrF kfrF cf]6f km/s km/s 7f]; j:t' k|ofu] u/L cg'¿k cfs[lt lvRg'xf];\ .
4. cfˆgf] bj' } xft hf]8]/ gd:sf/ ug{'xf;] \ . tL b'j} xTsn] fx¿ Ps csf;{ Fu cg¿' k

5g\ ls 5}gg\ < ;fyL;Fu 5nkmn ugx'{ f;] \ .
5. cfkm\gf] 3/df ePsf ?= 1 / ?= 2 sf b'Oc{ f]6f l;Ssfx¿ lngx' f];\ / pSt

l;Ssfx¿ cg'¿k 5g\ ls 5}gg\ < kl/jf/sf ;b:ox¿;uF 5nkmn ug{'xf;] \ .
kl/of]hgf sfo{

;a} ljBfyL{ pko'St ;dx" df ljefhg eO{ kT| o]s ;dx" n] cfcfˆgf] 3/ tyf
ljBfno j/k/ ePsf cg¿' k cfs[ltx¿ h:t}M l;Ssf, gf]6, ?dfn, lstfa,
O/]h/ cflb ;ªs\ ng u/L sIffdf kb| {zg ug{x' f;] \ .
pQ/
lzIfsnfO{ bv] fpg'xf];\ .

226 ul0ft, sIff &

kf7 16 7f;] j:tx' ¿

(Solid Objects)

16.0 kg' /jnfs] g (Review)

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

j:ts' f] gfd ;dtnLo cfsl[ tx¿ 7f]; cfsl[ tx¿
;nfOs{ f] a66\ f cfot if8\dv' f
8fO; ju{ 3g
cfO;lj|mdsf] aflx/L vfn] j[Q
8«d

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

16.1 6]6«fx8] g« (Tetrahedron)
ljm| ofsnfk 1

pko'St ;ªV\ ofdf ljBfyL{sf] ;dx" agfpgx' f];\ / kT| o]s
;dx" n] Ps Pscf]6f bfofsF f] lrqdf ePsf] h:t} 7f;] j:t'
lbgx' f;] \ . tL j:t'sf] cjnfs] g u/L tnsf k|Zgx¿sf af/d] f
;dx" df 5nkmn ug{x' f];\ M

-s_ s] o;sf ;a} lsgf/fx¿ a/fa/ 5g\ <
-v_ s] kT| os] ;tx ;dafx' lqe'h cfsf/sf 5g\ <
-u_ o;sf ;txx¿ sltcf]6f 5g\ <
-3_ o;df slt sltcf6] f lsgf/f / zLif{ laGb'x¿ 5g\ <
-ª_ of] lgoldt 7f]; j:t' jf clgoldt 7f]; j:t' s'g xf] <
;dx" df 5nkmn ul/;sk] l5 ;dx" sf] lgisif{ sIffdf k|:t't ug{'xf;] \ .

6]6f« x8] g« Pp6f lgoldt HofldtLo 7f;] cfs[lt xf] . o;sf k|Tos] ;txx¿

;dafx' lqe'haf6 ag]sf xG' 5g\ . o;df hDdf 4 cf]6f ;txx¿ 4 cf]6f
zLifl{ aGb' / 6 cf]6f lsgf/fx¿ x'G5g\ .

ul0ft, sIff & 227

16.1.1 66] f« x]8g« sf] vfj] |mf] gdg' f (Skeleton of Tetrahedron) lgdf0{ f
ljm| ofsnfk 2

pko'St ;ªV\ ofdf ljBfyL{sf] ;dx" agfpg'xf;] \ . kT| os]
;d"xn] 6 cf]6f a/fa/ gfksf l;Gsfx¿ / 4 6j' m| f cfn' jf
cGo g/d j:t'sf 6'jm| fx¿ lng'xf];\ . ca lrqdf b]vfP
h:t} u/L l;Gsfx¿ / cfns' f 6'j|mfx¿ hf]8\g'xf];\ . To;kl5
cjnfs] g u/L ;d"xdf ;fyLx¿;Fu 5nkmn ug{'xf;] \ / tnsf
k|Zgx¿sf] pQ/ vfH] gx' f;] \ .

-s_ s:tf] cfsl[ t aGof] <
-v_ sltcf6] f lsgf/fx¿ / sltcf6] f s'gfx¿ ag] <
;dx" 5nkmnaf6 cfPsf lgisif{ sIffdf k|:t't ug{x' f;] \ .

16.2 cS6fx8] «g (Octahedron)

ljm| ofsnfk 3
;dx" df bfofsF f] lrqdf lbOPsf] h:t} 7f]; j:t' lng'xf;] \ .
pSt 7f;] j:tn' fO{ cjnf]sg u/L tnsf k|Zgx¿sf af/]df
5nkmn ug{x' f;] \ M

-s_ s] o;sf kT| o]s lsgf/fx¿ a/fa/ 5g\ <
-v_ s] o;sf k|To]s ;tx ;dafx' lqe'h cfsf/sf 5g\ <
-u_ o;sf sltcf]6f ;txx¿ 5g\ <
-3_ o;df sltcf6] f lsgf/fx¿ / sltcf]6f zLif{ laGb'x¿

5g\ <
-ª_ of] 7f]; cfsl[ tsf] 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'
lqeh' cfsf/sf xG' 5g\ . o;df hDdf 8 cf]6f ;txx¿, 6 cf6] f zLif{
laGb'x¿ / 12 cf6] f lsgf/fx¿ x'G5g\ .

228 ul0ft, sIff &

16.2.1 cS6fx8] g« sf] vfj] m| f gdg' f (Skeleton of Octahedron ) lgdf{0f

ljm| ofsnfk 4

pko'St ;ª\Vofdf ljBfyL{sf] ;d"x agfpgxf];\ .
kT| os] ;d"xn] 12 cf]6f a/fa/ gfksf l;Gsfx¿ / 6
cf]6f cfns' f 6'jm| fx¿ jf cfn' h:t} j:ts' f 6'j|mfx¿
lng'xf;] \ ca lrqdf b]vfP h:t} u/L l;Gsfx¿ /
cfn'sf 6'jm| fx¿ hf]8\gx' f;] \ . To;kl5 cjnf]sg
u/L ;dx" df 5nkmn u/L tnsf k|Zgx¿sf] pQ/
vf]Hgx' f;] \ / sIffdf k:| t't ug{x' f];\ .

-s_ of] 7f]; cfsl[ tsf] gfd s] xf] <
-v_ o;sf sltcf6] f lsgf/fx¿ 5g\ <
-u_ o;df sltcf6] f zLifl{ aGb'x¿ 5g\ <
-3_ o;sf ;txx¿ sltcf]6f / s:tf s:tf 5g\ <

ljm| ofsnfk 5

sIffsf ;Dk0" f{ ljBfyL{ kfrF cf6] f ;d"xdf ljefhg eP/ a:g'xf;] \ .

kT| o]s ;d"xdf jm| dzM 3g (Cube), 6]6«fx8] «g(Tetrahedron),cS6fx8] «g (Octahedron),
8f8] s] fx]8«g (Dodecahedron) / cfOsf];fx]8g« (Icosahedron) sf 7f]; gdg' fx¿
cjnfs] g u/L tnsf kZ| gx¿sf af/d] f 5nkmn ug'{xf];\ .

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

ca ;a} ;d"xn] cfˆgf] ;d"xn] u0fgf u/]sf sg' fx¿sf] ;ªV\ of, ;txx¿sf] ;ªV\ of
/ lsgf/fx¿sf] ;ª\Vof tnsf] tflnsfdf eg'{xf]; . To;kl5 ltgLx¿sf] ;DaGwsf
af/]df 5nkmn u/L sIffdf k:| tt' ug'x{ f;] \ .

ul0ft, sIff & 229

j|m=;= 7f;] j:tx' ¿ zLif{ lsgf/fx¿sf] ;txx¿sf] V, E / F sf]
laGbx' ¿sf] ;ª\Vof (E) ;ªV\ of (F) ;DaGw
1 3g ;ª\Vof (V)
2 6]6«fx8] g«
3. cS6fx8] g«
4. 8f]8]sfx8] «g
5. cfOsf];fx8] g«

3g, 6]6f« x8] «g, cS6fx8] g« , 8f]8]sfx8] «g / cfOsf;] fx]8«gdf zLif{ laGb'x¿sf]
;ªV\ of (V) ;txx¿sf] ;ªV\ of (F) / lsgf/fx¿sf] ;ªV\ of (E) sf] ;DaGw
V – E + F = 2 xG' 5 .

pbfx/0f 1

Pp6f 66] «fx8] «gdf 4 cf]6f zLiflaGbx' ¿ / 6 cf6] f lsgf/fx¿ 5g\ eg] ;txx¿sf]
;ªV\ of kQf nufpg'xf];\ .
;dfwfg
oxfF 6]6«fx]8«gsf zLifl{ aGb'sf] ;ªV\ of (V) = 4
lsgf/fsf] ;ª\Vof (E) = 6
;txsf] ;ªV\ of (F) = ?
xfdLnfO{ yfxf 5,

V–E+F=2

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

F=4

230 ul0ft, sIff &

cEof; 16.1

1. tnsf jfSox¿ l7s jf al] 7s s] x'g, 5'6\ofpg'xf];\ M

-s_ 6]6f« x8] g« sf k|To]s ;tx ;dafx' lqeh' cfsf/sf xG' 5g\ .
-v_ 66] «fx]8«gsf ;a} lsgf/fx¿ a/fa/ xG' 5g\ .
-u_ 6]6«fx]8«gdf hDdf tLgcf]6f ;txx¿ x'G5g\ .
-3_ cS6fx8] g« sf k|Tos] ;tx ;dafx' lqeh' cfsf/sf xbF' }gg\ .
-ª_ cS6fx]8«gdf hDdf 4 cf6] f lsgf/fx¿ x'G5g\ .
2. tnsf kZ| gx¿sf] pQ/ nV] g'xf];\ M

-s_ 6]6f« x8] «g eg]sf] s] xf] <
-v_ 66] f« x8] «g / cS6fx8] «gsf sg' } bO' c{ f6] f km/s n]Vg'xf];\ .
-u_ 8f8] ]sfx]8«g / cfOsf;] fx8] g« sf ;tx, lsgf/f tyf sg' fsf] ;DaGw

hgfpg] ;q" n]Vgx' f];\ .
-3_ cS6fx]8g« sf] k|Tos] ;tx (Face) s:tf] cfsf/sf] xG' 5 <
-ª_ cS6fx8] g« df sltcf]6f zLiflaGbx' ¿ (Vertices) / lsgf/fx¿ (Edges)

x'G5g\ <
3. Pp6f 66] f« x8] «gdf lsgf/fx¿ (Edges) / ;txx¿ (Faces) sf] ;ªV\ of

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

;ª\Vof 6 xG' 5, kQf nufpgx' f];\ .

kl/of]hgf sfo{

h'; vfg] kfOk, 5j\ fnL, af;F tyf lgufnf,] 86kg] sf vfnL l/lkmn tyf
wfufsf] ko| f]u u//] ljleGg gfksf 3g, 66] f« x]8«g / cS6fx8] g« sf gd'gfx¿
lgdf{0f u/L sIffdf 5nkmn u/L k|bzg{ ugx'{ f;] \ .

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

ul0ft, sIff & 231

16.3 ;f]nL / an] gf (Cone and Cylinder)

16.3.1 ;f]nL (Cone)
ljm| ofsnfk 1

pko'St ;ª\Vofdf ;d"xdf a:g'xf];\ . ;a} ;d"xn] tn lbOPsf h:t} cfsf/sf Ps
Pscf]6f j:tx' ¿ lngx' f];\ . pSt j:t'x¿sf] cjnf]sg u/L tnsf kZ| gx¿sf] pQ/
;dx" 5nkmn u/L vfH] gx' f;] \ M

-s_ kT| os] j:t' s:tf cfsf/sf 5g\ <
-v_ k|To]s j:t'sf cfwf/x¿ s:tf cfsf/sf 5g\ <
-u_ k|To]s j:ts' f] ;tx s:tf] cfsf/sf] 5 <
-3_ k|To]s j:t'df sltcf6] f zLifl{ aGb'x¿ jf s'gfx¿ 5g\ u0fgf ugx{' f];\ .
dflysf j:t'df Pp6f sg' f jf zLifl{ aGb', Pp6f cfwf/ j[Qfsf/ / Pp6f aj|m ;tx
/x]sf 5g\ . oL 7f]; cfs[ltx¿ ;a} ;fn] L (Cone) x'g\ .

;f]nLsf u0' fx¿
Pp6f zLif{ laGb,' Pp6f ajm| ;tx / Pp6f j[Qfsf/ cfwf/
ePsf] 7f]; cfsl[ tnfO{ ;fn] L (Cone) elgG5 .
-s_ o;df Pp6f zLifl{ aGb' xG' 5 .
-v_ cfwf/ jQ[ fsf/ xG' 5 .
-u_ Pp6f jj|m ;tx /x]sf] xG' 5 .

232 ul0ft, sIff &

;f]nLsf vfj] m| f gd'gf (Skeleton Model of Cone) lgdf{0f

ljm| ofsnfk 2

df5f dfg]{ 9l8of, cfO;ljm| dsf] vf]n h:tf ;fn] Lsf vf]jm| f gdg' fx¿sf] ;r" L tof/
u/L sIffdf ;fyLx¿;Fu 5nkmn ug'x{ f;] \ .

pko'St ;ªV\ ofdf ljBfyLs{ f] ;dx" agfpgx' f;] \ . kT| o]s ;dx" n]
Ps Pscf]6f j[Qfsf/ 7f]; j:t' / Pp6f sfuh lng'xf];\ . kT| o]s
;d"xn] sfuhdfly jQ[ fsf/ 7f]; j:t' /fvL j[Q agfpg'xf;] \ .
s}+rLsf] ;xotfn] j[Qsf] aflx/L 3]/f sf6g\ 'xf;] \ . To;kl5 j[QnfO{
l7s laraf6 b'O{ k6s k6\ofpgx' f];\ .

A O
OB A
B

ca k6o\ fOPsf] sfuhnfO{ vfn] /] rf/ efudWo] Ps efu s}rF Ln] sf6/] x6fpg'xf;] \ /
afFsL /x]sf efunfO{ lrqdf h:t} u/L hf8] ]/ udn] 6f:F g' xf] . s:tf] cfs[lt aGof] <
o;df sltcf6] f zLif{laGb,' sltcf]6f s'gf / sltcf6] f jQ[ fsf/ ;tx 5g\, cjnf]sg
u/L ;dx" sf 5nkmn sIffdf k|bzg{ ug{x' f];\ .

ul0ft, sIff & 233

an] gf (Cylinder )
lj|mofsnfk 3

kT| os] ;dx" n] tn lbOPsf h:t} 7f]; j:t'x¿ lng'xf;] \ . pSt j:tx' ¿sf÷cfsl[ tx¿sf]
cjnfs] g u/L tnsf k|Zgsf] pQ/ ;dx" df 5nkmn u/L vf]Hgx' f;] \ M

-s_ j:t' s:tf cfsf/sf 5g\ <
-v_ oL j:ts' f cfwf/ s:tf cfsf/sf 5g\ <
-u_ s] oL j:t'nfO{ u'8fpg ;lsG5 <
-3_ oL j:td' f sltcf6] f / s:tf ;dtnLo ;txx¿ 5g,\ u0fgf ug{x' f];\ .
oxfF ;a} 7f;] cfsl[ tdf÷j:t'df b'Oc{ f]6f jQ[ fsf/ ;txx¿ 5g\ . oL ;a} a]ngfsf/
j:t' xg' \ .
a]ngfsf u'0fx¿
-s_ o;sf cfwf/x¿ jQ[ fsf/ xG' 5g\ .
-v_ o;df Pp6f jj|m ;tx x'G5 .
-u_ o;sf cfwf/x¿ cfk;df ;dfgfGt/ xG' 5g\ .

cfwf/x¿ jQ[ fsf/ / ;dfgfGt/ eO{ Pp6f jj|m ;tx ePsf 7f]; j:tn' fO{
an] gf elgG5 . cyjf an] gf Pp6f 7f]; j:t' xf,] h;sf cfwf/x¿ jQ[ fsf/ /
;dfgfGt/ tyf Pp6f jj|m ;tx x'G5 .

16.3.2 a]ngfsf vfj] |mf gd'gf (Skeleton Model of Cylinder) lgdf0{ f
lj|mofsnfk 4

3/, ljBfno, ;8s lsgf/df cfkmn" ] bv] ]sf bfofsF f] lrqdf ePsf] h:t}
a]ngfsf vf]j|mf gdg' fsf] ;"rL tof/ u/L 5nkmn ug'x{ f];\ .
kT| o]s ;dx" n] Ps Pscf]6f cfotfsf/ sf8a{ f8] { kk] / lng'xf];\ . lrqdf
h:t} u/L cfotfsf/ nDafO;uF a/fa/ kl/lw ePsf pq} bO' c{ f6] f j[Qx¿

234 ul0ft, sIff &

lng'xf];\ . lrqdf h:t} u/L cfotfsf/ sf8{af8] { kk] /nfO{ pq} bO' c{ f6] f j[Qx¿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 gx' f;] \ .
s:tf] cfsf/ aGof] <
ags] f] cfs[ltsf] gfd s] xf] < o;df sltcf]6f jQ[ fsf/ ;txx¿ / sltcf]6f
zLif{laGb'x¿ 5g\ < cjnfs] g u/L ;dx" df 5nkmn ug{x' f;] \ / sIffdf k:| t't ug{'xf];\ .

cEof; 16.2
1. tnsf kZ| gx¿sf] pQ/ lbgx' f];\ M

-s_ ;fn] L egs] f] s] xf] < ;fn] Lsf sg' } b'Oc{ f]6f u'0f n]Vgx' f];\ .
-v_ a]ngf eg]sf] s] xf] < s'g} bO' c{ f6] f u'0f n]Vg'xf];\ .
-u_ a]ngf / ;fn] Ldf ePsf Pp6f ;dfgtf / Pp6f km/s nV] gx' f;] \ .
-3_ a]ngf / ;fn] Lsf] Pp6f Pp6f gdg' f lrq agfpg'xf;] \ .
2. ;f]nLsf] ;tx / cfwf/ s:tf s:tf xG' 5g\, n]Vg'xf];\ .
3. s'g} kfrF cf6] f 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{

tkfOsF{ f] 3/df ePsf jf 3/df ko| fu] ug]{ kfFr kfFrcf6] f an] gfsf/ /
;f]nL cfsf/sf j:t'x¿ vf]hL u/L sIffdf k:| tt' ug'x{ f];\ .

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

ul0ft, sIff & 235

kf7 17 lgbz]{ fª\sx¿

(Coordinates)

17.0 kg' /jnf]sg

;fyLx¿;Fu ldn/] lrqdf b]vfP h:t} u/L sfuhsf 6'j|mfx¿df X - cIf / Y- cIfsf
;ªV\ of /]vf agfpgx' f];\ .

ca b'O{ ;d"x (A / B) df afFl8P/ lgb{z] fªs\ vn] v]Ng'xf;] \ .
vN] g] tl/sf

-s_ ;jk{ y| d ;ª\Vof kQLx¿nfO{ lrqdf b]vfP h:t} cfk;df nDa xg' ] u/L rp/df
/fVg'xf];\ . ;a} hgf bO' {cf6] f ;d"xdf ljefhg eO{ cfd'Gg] ;fdG' g] x'g] u/L
a:gx' f];\ .

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

-v_ klxnf ;dx" A sf ;fyLn] ;dx" B sf ;fyLnfO{ sg' } laGbs' f] lgb{]zfªs\ sf
cfwf/df pSt laGb'df pleg eGg'xf];\ . l7s :yfgdf pleg g;s]df aflxl/g'
k5{ eGg] hfgsf/L u/fpgx' f];\ .

-u_ ca, ;dx" B sf ;fyLn] ;dx" A sf ;fyLnfO{ s'g} laGbs' f] lgb{z] fª\ssf cfwf/df
pSt laGbd' f pleg eGgx' f];\ . n]vflrqsf] cfˆgf] lgb]z{ fª\sdf plePsf] ;fyLn]

236 ul0ft, sIff &

g} ;dx" A sf] csf]{ ;fyLnfO{ g= -v_ df h:t} u/L s'g} csf]{ gofF laGbs' f]
lgb{z] fªs\ eGg] 5 / ;f]xLcg;' f/ plegk' 5{ .

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

-ª_ b'j} ;dx" x¿dWo] h'g ;d"xsf w/} ;fyLx¿ lgbz]{ fª\scg';f/ l7s laGb'df
pleg ;s] pxL ;d"xsf] lht x'G5 .

17.1 n]vflrqdf laGb'sf] lgb{z] fª\s

ljm| ofsnfk 1

pko'St ;ªV\ ofdf ljBfyLs{ f] ;dx" Y A
agfpgx' f];\ / ;uF s} f] u|fkm ePsf] lrqsf] C
cWoog u/L tnsf kZ| gx¿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 kU' g slt
PsfO afofF uP/ slt PsfO tn

hfgk' 5{ <

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

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

ljm| ofsnfk 2

;dx" df a;L ;Fus} f] lrqdf 5nkmn u/L tnsf k|Zgx¿sf] pQ/ vfH] g'xf];\ / sIffdf
k|:t't ugx{' f];\ M

-s_ laGb' A s'g rty' fz{F df k5{ <

ul0ft, sIff & 237

-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]{zfªs\ slt xfn] f <

-ª_ laGb' B sf] lgb{]zfª\s slt xfn] f < X'

-r_ ca laGb' C sf] lgbz{] fª\s slt
xfn] f <

Y'

s'g} klg laGbs' f] lgb]{zfªs\ n] To; laGbs' f] cjl:yltnfO{ hgfpbF f

X – lgbz]{ fªs\ n] pb\ud laGbe' Gbf slt PsfO bfofF jf afofF eGg] ae' mfp5F .
To:t} Y – lgb{]zfªs\ n] pb\ud laGbe' Gbf slt PsfO dfly jf tn eGg]
a'emfp5F .

17.2 nv] flrqdf lbOPsf laGbx' ¿sf] cªs\ g (Plotting the Given Points

in the Graph)

ljm| ofsnfk 3 Y A (6, 5)
B (–3, 5)

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

;jk{ y| d sIffsf ljBfyL{ X' O X
;d"xdf ljefhg eO{ cfcfˆgf] Y' D (4, –4)

uf| kmsfkLdf ;uF }sf] lrqdf

bv] fP h:t} u/L X – cIf / C (–4, –3)

Y – cIf hgfpg] ;ª\Vof /]vfx¿

agfpgx' f];\ .

238 ul0ft, sIff &

-s_ A(6, 5) nfO{ nv] flrqdf cª\sg ug{ s] ug{k' nf{, 5nkmn ugx'{ f;] \ .

A df k'Ug, X – cIfdf pb\ud laGb'bl] v 6 PsfO bfofF hfgk' g{] x'G5 . To;kl5
ToxL laGb'af6 Y– cIfdf 5 PsfO dfly uO{ A(6, 5) cª\sg ug'{k5{ .

-v_ To;} u/L j|mdzM laGb' B (-3, 5), C (–4, –3) / D(4, –4) nfO{ s;/L nv] flrqdf
cª\sg ug{ ;lsG5 xf]nf < ;fyL;Fu klg 5nkmn ug'{xf;] \ .

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

pbfx/0f 1

laGbx' ¿ K (3, 4), L(–3, 4), M(3, –5) / N Pp6f cfotsf zLif{ laGbx' ¿ x'g\ eg,]
-s_ lbOPsf laGb'x¿nfO{ nv] flrqdf cª\sg ug'x{ f];\ .
-v_ laGb' N sf] lgbz{] fª\s kQf nufpgx' f;] \ .
-u_ laGb' K / L larsf] b'/L kQf nufpg'xf;] \ .
;dfwfg

-s_ lbOPsf] laGbx' ¿ K (3,4), Y K (3, 4)
L (-3,4), M(3,-5) nfO{ j|mdzM L (–3, 4)
cª\sg u/L ;Fus} f] nv] flrqdf

bv] fOPsf] 5 .

-v_ laGb' N df kU' g X- cIfdf X' OX
pbu\ d laGb'af6 3 PsfO afofF uO{ M (3, –5)
ToxLFaf6 5 PsfO tn hfg'kb5{ . N
t;y{ N sf] lgb]{zfªs\ (–3, –5) Y'
x'G5 .

-u_ laGb' K b]lv laGb' L larsf] sf]7f
uGtL ubf{ 6 PsfO 5 .

t;y{ K / L aLrsf] b/' L (KL) = 6
PsfO xG' 5 .

ul0ft, sIff & 239

cEof; 17

1. lrqdf lbOPsf HofldtLo cfsl[ tx¿sf zLif{ laGbx' ¿sf lgb{]zfªs\ x¿ 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¿ larsf] b/' L kQf nufpgx' f;] \ .

3. tn lbOPsf kT| os] laGb'nfO{ nv] flrqdf cª\sg ug'{xf;] \ .

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

4. tnsf kT| os] laGbn' fO{ nv] flrq agfO{ cªsg ug{'xf;] \ . kT| o]s laGbn' fO{ j|mdzM
hf8] \gx' f;] \ . o;/L hf]8b\ f aGg] cfs[ltsf] gfd klg n]Vg'xf];\ .

-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{ laGb'x¿ xg' \ eg]
-s_ lbOPsf laGbx' ¿nfO{ nv] flrqdf cªs\ g ug'x{ f;] \ .
-v_ AB sf] nDafO slt xf]nf <
-u_ CD sf] nDafO slt xf]nf <

240 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 laGb'nfO{ n]vflrqdf cª\sg ugx{' f;] \ .
-v_ laGb' M sf] lgb{]zfª\s nV] g'xf;] \ .
-u_ laGb' JK sf] nDafO slt xfn] f <

kl/ofh] gf sfo{
Pp6f 7'nf] ;fOhsf] uf| kmk]k/df X – cIf / Y – cIf agfpg'xf;] \ . pSt
uf| kmdf Pp6f lrq agfpgx' f];\ . pSt lrqsf] cjnfs] g u/L s'g}
kfFrcf]6f laGbx' ¿sf lgb]z{ fªs\ sf] n]Vgx' f];\ / sIffdf k|:tt' ug'x{ f];\ .

pQ/
lzIfsnfO{ b]vfpgx' f;] \ .

ul0ft, sIff & 241

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

(Symmetry and Tessellation)

18.0 k'g/jnf]sg (Review)

tn lbOPsf lrqx¿sf] cjnfs] g u/L ;d"xdf ;fyLx¿;Fu 5nkmn u/L ;fl] wPsf
kZ| gx¿sf] pQ/ vfH] g'xf];\ M

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

Figures) xg' \ 56' \ofpgx' f];\ .
-u_ s] dflysf lrqx¿nfO{ 180° sf]0fdf 3d' fpFbf klg p:t} b]lvG5g\ <

● a/fa/ efudf afF8g\ ;lsg] lrqnfO{ ;dldtLo lrq elgG5 .
● s'g} klg lrqdf hg' /]vfaf6 lrqnfO{ b'O{ 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. /]vf ;dldlt (Line Symmetry)
ljm| ofsnfk 1

;a} ljBfyLn{ ] lbOPsf cfsl[ tx¿sf] 6«;] ugx'{ f];\ M

lrq -s_ lrq -v_ lrq -u_

242 ul0ft, sIff &

cfkmn" ] 6;]« u/]sf lrqnfO{ 86 /v] f -;dldltsf] cIf_ bv] L a/fa/ efudf k6o\ fpgx' f];\ .
-s_ lrq -s_ nfO{ slt tl/sfn] bO' { a/fa/ efu x'g] u/L k6\ofpg ;lsof] <
-v_ lrq -v_ nfO{ slt tl/sfn] a/fa/ efu xg' ] u/L k6\ofpg ;lsof] <
-u_ lrq -u_ nfO{ slt tl/sfn] a/fa/ efu xg' ] u/L k6\ofpg ;lsof] <

cfˆgf] aG] rsf ;fyLx¿;uF 5nkmn ug'x{ f;] \ .
● lrq -s_ / lrq -v_ nfO{ 1 tl/sfn] k6o\ fpg ;lsG5 . t;y{ o;df /v] Lo
;dldltsf] cIf Pp6f dfq 5 .
● lrq -u_ nfO{ 2 tl/sfn] k6o\ fpg ;lsG5 . t;y{ o;df /]vLo
;dldltsf] cIf 2 cf]6f 5g\ .

ljm| ofsnfk 2

k|To]s ljBfyL{sf] Pp6f sfuhsf] kfgf lnO{ To;nfO{ lar efuaf6 lrqdf
bv] fP h:t} u/L k6o\ fpg'xf];\ . k6\ofOPsf] efunfO{ oyfjt\ /fvL kg' M csf]{ lt/af6
Ps k6s k6\ofpgx' f];\ .

-s_ -v_ -u_ -3_ -ª_

ca t;] f{] lrqdf lbOPsf] cfsl[ t agfpgx' f];\ . To;kl5 -3_ df h:t} u/L To;sf]
aflx/L 3/] f s}rF L]n] sf6\g'xf;] \ / k6o\ fOPsf] efunfO{ vfN] gx' f];\ .

o;/L ag]sf cfsl[ tdf sltcf]6f ;dldlt /v] fx¿ (Lines of Symmetry) ag],

5nkmn u/L kQf nufpgx' f];\ .

kT| os] lrqnfO{ b'O{ a/fa/ efudf afF8\g] 86 /]vf (Dot Line) nfO{ ;dldltsf]

cIf (Axis of Symmetry) elgG5 . o;nfO{ csf]{ zAbdf P]gf /v] f (Mirror
Line) klg eGg] ul/G5 .

ul0ft, sIff & 243

pbfx/0f 1

tn /]vf ;dldltsf] cIf / cfwf lrq lbOPsf] 5 . o;nfO{ k/" f ug{'xf;] \ . /v] Lo
;dldltsf] cIfsf] ;ª\Vof klg kQf nufpgx' f;] \ .
;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} ljBfyL{ pko'St ;d"xdf ljefhg eO{ k|To]s ;dx" n] lbOPsf] lrqnfO{
kf/bzL{ Knfl:6sdf 6;]« ugx'{ f;] \ .

A

O

B

dflysf] lrqdf l7s ldNg] u/L sG] b| O df k]lG;nsf] 6K' kfn] lyr]/ 6;]« u/]sf] lrqnfO{
lj:tf/} 3d' fpgx' f;] \ .

o;/L 3'dfpbF f,

● slt l8uL| sf] sf0] fdf 3'dfpbF f lrq -cfs[lt_ s]Gb|b]lv a/fa/ b'/Ldf t/
ljk/Lt lbzfdf cfOkU' 5 <

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

244 ul0ft, sIff &