गणित कक्षा ३

kf7 & leGg

k9g\ x' f];\ / 5nkmn ug'x{ f;] \ M

/df kNnf] 3/sf] ;fyLsxfF uPsL lyOg\ . ;fyLsL cfdfn] kfk8 kf]n]sf]
b]lvg\ .

cfdf kfk8 kfk8 t Pp6f dfq 5 . To;f] eP d
vfg] Û ltdLx¿nfO{ cfwf cfwf lbG5' x} .

d klg vfg] Û

d}n] cfwf kfPF . l;ª\uf] cfwf
d}n] klg cfwf kfPF .

s'g} l;ª\uf] j:tn' fO{ a/fa/ 6j' m| fx¿df efu nufP/
6j' |mfnfO{ hgfpg leGgsf] ko| f]u ul/G5 .

d}n] kfPsf] cfwfnfO{ leGgsf ¿kdf s;/L n]Vg] <

! cz+ Pp6f l;ª\uf] j:tn' fO{ hlt a/fa/ 6'j|mfx¿df
@ x/ afFl8G5 . tL a/fa/ 6'jm| fx¿sf] hDdf ;ª\VofnfO{

cfwf x/ elgG5 . oxfF x/ @ xf] .

Pp6f l;ª\uf] j:ts' f] s]xL efu ae' mfpg] ;ª\VofnfO{
c+z elgG5 . oxfF ! 6j' m| f kfPsfn] c+z ! xf] .

d/] f] ul0ft, sIff # 45

j;Gtn] Pp6f /f]6L b'O{ a/fa/ efu nufO{ cfwf /f6] L alxgLnfO{
lbP . alxgLn] /f]6L vfg dflggg\ . alxgLnfO{ lbPsf] /f6] L klg
cfkm‘n} ] vfP . pgn] /f6] Lsf] slt efu vfP <

b'O{cf]6f cfwf

! @ ≠ ! -l;ª\uf]_
@

oxfF @ efux¿df b'j} cfkm¹n] vfPsfn]

leGgdf x/ / c@+z a/fa/ eof] . ePsfn] ! n]lvG5 .
n]Vbf eof] . of] l;ª\uf]
@

tn lbOPsf l;ª\ufn] fO{ /v] f tfgL cfwf ugx{' f;] \ M

tn lbOPsf leGgn] hgfpg] efudf /ª nufpg'xf];\ M

@ !@!
@ @@@

46 d/] f] ul0ft, sIff #

k9g\ x' f];\ / 5nkmn ugx'{ f];\ M
Pp6f /f]6L rf/ hgfn] a/fa/ afF8]/ vfcf¹ .

$ rf/ rf}yfO
$

;lagf wg cfn] nvg

Ps rf}yfO Ps rf}yfO Ps rf}yfO Ps rf}yfO
!
!!! $
$ $ $
;lagf / wgn] u/L hDdf slt efu vfP <

$ efudf @ efu vfP .

;lagf wg b'O{ rf}yfO @ b'O{ rf}yfOnfO{
cfn] nvg $ cfwf klg elgG5 .

;lagf, wg / cfn]n] u/L hDdf slt efu vfP <

;lagf wg $ efudf # efu vfP .
cfn] nvg tLg rf}yfO #

$

d]/f] ul0ft, sIff # 47

tn lbOPsf lrqx¿df 5fof kfl/Psf] efu x/] L leGgdf nV] g'xf;] \ M

!
@

tn lbOPsf leGgnfO{ lrqdf 5fof kf/L bv] fpg'xf;] \ M

!!
@$
#@
$$

$ @
$ @

48 d/] f] ul0ft, sIff #

pldn{ fn] Pp6f nfdf] l/ag a/fa/ efudf sf6/] cfkm" / cfkm\gf
bO' { 5f/] Lx¿ piff / dlgtfnfO{ aflF 8g\ . kT| os] n] slt kfP <

a/fa/ efu nufpgx' f];\ x} Û cfxf Û gofF l/ag nufpg] Û

piff pld{nf dlgtf

pld{nfn] slt kfOg\ < # efudf ! efu <

# efusf] ! efunfO{
Ps ltxfO klg elgG5 .

pld{nf

! -Ps ltxfO_
#
# efudf @ efu <
piff / dlgtfn] u/L

hDdf slt kfP <

piff dlgtf

@ -b'O{ ltxfO_
#

d]/f] ul0ft, sIff # 49

hf8] f ldnfpgx' f];\ M

! bO' { ltxfO !
@ $
! !
$ cfwf #
!
# tLg rf}yfO @
$ @
#
! #
# Ps rf}yfO $
@
# Ps ltxfO

lbOPsf] leGgnfO{ 5fof kf/L b]vfpgx' f];\ M

! cfwf $! Ps rf}yfO @# b'O{ ltxfO
@

$# tLg rf}yfO ! Ps rf}yfO @ b'O{ rf}yfO
$ $
50 d]/f] ul0ft, sIff #

tn lbOPsf leGgx¿nfO{ hgfpg] lrq agfO{ 5fof kf/L bv] fpgx' f;] \ M

!!
@$
!@
##
#@
$$
#$
#$

d]/f] ul0ft, sIff # 51

tn lbOPsf k|Tos] lrqdf /ª\ufOPsf] efunfO{ hgfpg] leGg
n]Vg'xf;] \ . - _ df '>' cyjf '<' lrxg\ n]Vgx' f;] \ M

lrq leGg

!
@
! ! > !
@ $

$

52 d]/f] ul0ft, sIff #

lbOPsf leGgx¿nfO{ lrq x/] L tn' gf ugx{' f];\ / '>' jf '<' lrx\g
ko| f]u u/L nV] gx' f];\ M

# !! @
$ $# #

! @# @
@ @$ $

lbOPsf leGgx¿nfO{ lrq x]/L t'ngf ug{x' f;] \ / '>' jf '<' lrxg\
ko| f]u u/L n]Vg'xf];\ M

!! !@
@$ #$
#@
$# !#
@$

d/] f] ul0ft, sIff # 53

5nkmn ugx{' f;] \ / nV] g'xf];\ M

!= lrqdf sltcf]6f kmnkm'nx¿ 5g\ <
@= hDdf ;G' tnf sltcf]6f 5g\ <
#= ;G' tnfsf] ;ª\VofnfO{ leGgdf n]Vg'xf];\ .
$= s]/fsf] ;ª\VofnfO{ leGgdf n]Vg'xf];\ .

tn lbOPsf] leGgsf] cfwf/df pbfx/0fdf lbOPh:t} u/L 3]/f
nufpgx' f];\ M

cfwf !
@

Ps ltxfO ! b'O{ ltxfO @
# #

tLg rf}yfO # cfwf !
$ @

54 d]/f] ul0ft, sIff #

tn lbOPsf] leGgsf cfwf/df 3]/f nufpgx' f];\ M

Ps ltxfO ! b'O{ ltxfO @ cfwf !
# # @

cfwf ! Ps rf}yfO ! b'O{ rf}yfO @
@ $ $

bO' { rf}yfO @ tLg rf}yfO # cfwf !
$ $ @

d]/f] ul0ft, sIff # 55

kf7 * nDafO !

j:t'sf] nDafO

?n/sf] k|ofu] u/L tnsf j:ts' f lrqx¿ slt nfdf 5g,\
gfKgx' f;] \ / sIffdf k:| tt' ug{x' f];\ M

-s_ -v_

-u_ -3_

-ª_ -r_

56 d/] f] ul0ft, sIff #

5nkmn ug{x' f];\ M
tnsf] lrq j:ts' f] nDafO gfKg k|of]u ul/g] 6]ksf] xf] . 5f]6f] j:ts' f] nDafOsf] gfk
;]lG6ld6/ PsfO / nfdf] j:t'sf] nDafOsf] gfk ld6/ PsfOdf gflkG5 .
sIffsf]7fsf] nDafO, 3/sf] prfO, 9f]sfsf] prfO cflb gfKg ld6/ PsfOsf]
ko| f]u ul/G5 .

Ps ld6/df !)) ;]lG6ld6/ x'G5 .
! ld6/ ≠ !)) ;]lG6ld6/

sIffsf]7fdf cfkm" a:g] a]Gr slt nfdf] xf]nf <
a]Grsf] nDafO ;]lG6ld6/ PsfOdf gfKg ;lsPnf t <
;lsG5 t/ ld6/ PsfOdf
gfKg pkoS' t xG' 5 .

sIffsf]7fdf em'G8\ofOPsf]
kf6L slt nfdf] xf]nf <

o;sf] prfO slt xf]nf <

of] unr¹ f @ ld= %)
;]=ld=nfdf] 5 .

d]/f] ul0ft, sIff # 57

tkfOs‘{ f] sIffsf7] fdf ePsf tnsf j:tx' ¿sf] nDafO gfKgx' f];\ M
kf6L 6]a'n

a]Gr s;' L{sf] a:g] efu

tkfO‘s{ f] 3/, sf]7f / 3/sf] ;'Tg] sf7] fdf ePsf j:t'x¿sf]
nDafO gfKgx' f;] \ M

vf6sf] nDafO ;T' g] sf]7fsf] nDafO

3/sf] nDafO 6]a'nsf] ;txsf] nDafO


l;/ssf] nDafO b/fhsf] nDafO

cfkmg\ f] jl/kl/ ePsf s'g} bO' c{ f6] f j:t'x¿sf] gfk ;l] G6ld6/df
/ cGo bO' {cf]6f j:tx' ¿sf] gfk ld6/df gfk/] n]Vg'xf];\ M

58 d]/f] ul0ft, sIff #

j:t'x¿sf] nDafOsf] cg'dfg ug'x{ f;] \ M
of] bfofFk6\l6sf] lrq sfkLsf] xf] .
sfkLsf] ;Fu}sf b'O{cf]6f lsgf/fsf] gfk
km/s km/s 5 .
7f8f] efusf] nDafO w]/} xf]nf ls t];f]{ efusf] <
o;sf] 7f8f] efusf] nDafO slt ;]lG6ld6/
xf]nf <
o;sf] t];f]{ efusf] nDafO slt ;]lG6ld6/ xf]nf <

bfofFk6\l6sf] lrq sndsf] xf] .
of] slt nfdf] xf]nf <
of] snd !@ ;]=ld=nfdf] xf]nf t <
of] snd % ;]=ld=nfdf] xf]nf <
of] snd @) ;]=ld=nfdf] xf]nf <

of] rs6L / 6]a'nsf] lrq xf] .
rs6L slt nfdf] xf]nf <
6]a'nsf] ;txsf] lsgf/f slt nfdf] xf]nf <
6]an' / rs6Ldf sg' sf] nDafO w]/} xf]nf <

d]/f] ul0ft, sIff # 59

tkfO;‘{ Fu ePsf tnsf j:tx' ¿sf] gfk cgd' fg u/L l7s (√)
lrgf] nufpgx' f;] \ M

$ ;]=ld= & ;]=ld= !@ ;]=ld= !^ ;]=ld=


% ;]=ld= !) ;]=ld= @) ;]=ld= #) ;]=ld=



@) ;]=ld= #) ;]=ld= %) ;]=ld= *) ;]=ld=



tn lbOPsf j:tx' ¿sf] nDafO cg'dfg u/L gfk]/ hfFRg'xf;] \ M

j:t' cgd' flgt jf:tljs km/s
nDafO nDafO

tkfO‘{sf] cfkmg\ f] prfO cg'dfg ug'{xf;] \ / gfk/] jf:tljs prfO

kQf nufpg'xf];\ M

cg'dflgt prfO jf:tljs prfO

60 d]/f] ul0ft, sIff #

nDafOsf] cg'dfg ugx'{ f;] \ M

6f9fsf] b'/L gfKg ;]lG6ld6/ PsfOdf sl7g
x'G5 . ld6/ PsfOsf] ko| f]u ug'{k5{ .

dflysf] lrqdf ljBfyL{ / 3/larsf] b/' L slt xf]nf <
ljBfnoaf6 ufO{x¿ slt 6f9f xf]nfg\ <

o; 6]an' sf] dflysf] ;txsf]
nDafO @ ld6/ 5 .
csf]{lt/sf] nDafO slt xf]nf <

d]/f] ul0ft, sIff # 61

knªsf] nDafO slt 5 <

tn lbOPsf j:t'x¿sf] nDafO cgd' fg ugx{' f;] \ M

j:t' cg'dflgt jf:tljs km/s
gfk gfk
cfkm\gf] sIffsf]7fsf] nDafO
ljBfno ejgsf] nDafO
ljBfno v]n d}bfgsf] nDafO
cfkm" a:g] a]Grsf] nDafO

tn lbOPsf j:tx' ¿sf] nDafO cg'dfg ug{x' f];\ / gfk lnP/
hfrF ug{x' f;] \ M

j:t' cg'dflgt jf:tljs km/s
gfk gfk
knªsf] nDafO
;T' g] sf]7fsf] nDafO
sf]7fsf] b/fhsf] nDafO
efG;fsf]7fsf] nDafO

62 d/] f] ul0ft, sIff #

xfdf| ] ;db' fo

x/] f‘,} dn} ] slt l;s]F <

!= !, $ / # af6 aGg] sg' } tLgcf]6f ;ªV\ ofx¿ n]Vgx' f;] \ / tL
;ªV\ ofx¿nfO{ a9\bf] jm| d / 36b\ f] jm| ddf nV] g'xf];\ M

;ª\Vofx¿ M

a9\bf] jm| d M

36\bf] jm| d M

@= lbOPsf leGgx¿nfO{ lrqdf /ª e//] bv] fpgx' f;] \ M

cfwf Ps rf}yfO tLg rf}yfO b'O{ rf}yfO

#= cfkmg\ f] j/k/ ePsf j:tx' ¿sf] nDafO cg'dfg u/L tn lbOPsf

gfkx¿;uF ldNbf] gfk ePsf Ps Pscf6] f j:ts' f] gfd n]Vgx' f];\

M ldlnld6/ 1 ;]lG6ld6/ 1 ld6/

1

$= sg' } Pp6f :yfgLo ;ª\Vofª\sg k4ltdf ! bl] v !) ;Ddsf
;ªV\ ofx¿ nV] g'xf;] \ M

d]/f] ul0ft, sIff # 63



%= Pp6f j:tn' fO{ a/fa/ x'g] u/L 6j' m| fdf efu nufOPsf] 5 .
/ª\ufOPsf 6j' |mfx¿n] hgfpg] leGgx¿ nv] L t'ngf ug'{xf;] \ M

^= tn lbOPsf leGgx¿sf cfwf/df 3/] f nufpgx' f];\ M

!@ !
@# #

cleefjssf] b:tvt
lzIfssf] b:tvt

64 d/] f] ul0ft, sIff #

kf7 ( d/] f] l;h{gf $
uLt ufcf}‘ M
/v] fx¿

/]vf, sf0] f, lqeh' , rte' {'h

lstfa sfkL lemsf¹ ;fyL vf]nf¹ ca emf]nf .
/]vf;Fu v]nf¹ ;fyL s] s] aG5 xf]nf ..

Pscfk;df 5nkmn ub}{ hfg] kl5 5f]8 .
bO' { l;wf /]vf hf]8\bf aG5 ;fyL sf]0f ..

sf]0fdf lrx\g nufcf¹ ;fyL gu/f¹ x} a]/ .
tLg e'hf hf]8]F d}n] lqeh' aGof] x]/ ..

rf/lt/ ws}{ wsf{ sf]/f¹ ca ;fyL .
rf/ eh' ] cfs[lt t aG5 lgs} vfF6L ..

o:tf] aGb cfs[ltnfO{ s] eG5g\ xf] ;'g .
rte' '{h eG5g\ ;fyL dgdg} u'g ..

b'O{df sf]0f, tLg lqeh' , rf/ rt'e{'h .
oL t ;a} HofldtLo cfs[lt x'g\ a'em ..

d]/f] ul0ft, sIff # 65

yf]Knfx¿ hf]8/] cfs[lt k"/f ug'{xf];\ / lqeh' df /ftf,] j[Qdf lgnf]
/ rte' {h' df kx]Fnf] /ª eg'{xf;] \ M

66 d]/f] ul0ft, sIff #

6]a'n jl/kl/ en] f xg' 'xf;] \ . rf/ sg' f / l;wf lsgf/fx¿ 5'g'xf];\ M

xftdf ;nfO{sf] a6\6f lnO{ lsgf/fx¿ 5'g'xf];\ M
?n/ k|ofu] u/L ;nfOs{ f] a6\6fsf lsgf/fx¿ gfKg'xf];\ M

% ;]lG6ld6/

d/] f] ul0ft, sIff # 67

?n/ k|of]u u/L ;nfOs{ f] a6\6fsf lsgf/fx¿ gfKgx' f;] \ M
;]lG6ld6/

;]lG6ld6/

68 d/] f] ul0ft, sIff #

l;;fsndsf] 6'Kkf] sfuhdf
/fv]/ p7fpFbf aGg] yf]Knf] - _ laGb' xf] .
r/fsf] lrqdf cfFvf a'emfpg] u/L yf]Knf lbgx' f;] \ M

df5fsf] lrqdf cfvF f a'emfpg] u/L yfK] nf lbgx' f;] \ M

yf]Knfx¿ k|of]u u/L cfs[lt agfpg'xf;] \ M

d]/f] ul0ft, sIff # 69

aGb cfs[ltleq s'g} bO' { 7fpdF f yf]Knf lbgx' f];\ M

?n/sf] ;xfotfn] lbOPsf bO' { laGb' -yf]Knf_ x¿ hf]8L /]vfv08x¿
lvRgx' f];\ M

D

PQ

A

BC

?n/ ko| fu] u/L dflysf /]vfv08x¿sf] gfk lng'xf];\ / n]Vg'xf;] \ M

P b]lv Q ;Dd hf]8\g] /]vfv08nfO{ PQ n] hgfpFbf,

PQ Ö ;]=ld=

A b]lv B ;Dd hf]8\g] /]vfv08nfO{ AB n] hgfpFbf,

AB Ö ;]=ld=

C b]lv D ;Dd hf]8\g] /]vfv08nfO{ CD n] hgfpFbf,

CD Ö ;]=ld=

70 d/] f] ul0ft, sIff #

l;;fsndsf] cg'dflgt gfkM ;]=ld=

jf:tljs gfkM ;]=ld=

lbOPsf /]vfv08x¿sf] nDafO cg'dfg u/L n]Vg'xf;] \ . To;kl5
?n/ ko| fu] u/L jf:tljs gfk nV] g'xf;] \ M

-s_ cg'dflgt gfk ;]lG6ld6/
jf:tljs gfk ;]lG6ld6/

-v_ cgd' flgt gfk ;]lG6ld6/
jf:tljs gfk ;]lG6ld6/

-u_ cgd' flgt gfk ;]lG6ld6/

jf:tljs gfk ;]lG6ld6/

-3_ cgd' flgt gfk ;]lG6ld6/
jf:tljs gfk ;]lG6ld6/

d/] f] ul0ft, sIff # 71

lbOPsf gfkcg;' f/sf /]vfv08x¿ lvrL gfd nV] g'xf;] \ M

# ;]=ld= C D

of] /]vfv08 CD xf] .

@ ;]=ld=

$ ;]=ld=

^ ;]=ld=

* ;]=ld=

!) ;]=ld=

% ;]=ld=

& ;]=ld=

72 d]/f] ul0ft, sIff #

tn lrqdf lbOPsf] h:t}u/L ?n/sf] ;xfotfn] ! ;uF !, @ ;Fu @,
# ;Fu # ub}{ !) ;uF !) hf8] g\ x' f];\ M

!
@
#
$
%
^
&
*
(
!)

! @ # $ % ^ & * ( !)

!)
!(
@*
#&
$^
%%
^$
&#
*@
(!
!) ! @ # $ % ^ & * ( !)

d/] f] ul0ft, sIff # 73

kf7 !) sf]0fx¿

gf]j{'sf] n6\7L efFlrbf s:tf] cfsl[ t aGof], 5nkmn ugx'{ f];\ M

efFlrPsf] n6\7Ldf
sltcf]6f s'gf 5g\ <

o:tf] cfs[ltnfO{ sf]0f elgG5 .
sxfF sxfF sf0] fx¿ ag]sf 5g\ < 5nkmn ugx'{ f;] \ .

11 12 1
10 2
93
84

765

tkfO{‘x¿sf] jl/kl/ sxfF sxfF sf]0f ags] f bV] ge' Psf] 5 < sg' }
rf/cf6] f cj:yf nV] g'xf;] \ M

!= @=
#= $=

74 d/] f] ul0ft, sIff #

cWoog ug{'xf;] \ M

tnsf laGbx' ¿nfO{ A b]lv B ;Dd ?n/ ko| f]u u/L hf]8f¹ . To;} u/L B
b]lv C ;Dd klg hf]8f¹ . o;/L ag]sf] cfs[ltnfO{ sf]0f elgG5 .

AA

B CB C

/]vfv08 AB / /]vfv08 BC laGb' B df ldn]sf 5g\ . laGb' B nfO{ sf]0fsf]
zLif{laGb' elgG5 .

laGb'x¿ A / C clGtd laGbx' ¿ xg' \ . sf]0fsf] gfd n]Vbf, sf]0fsf]
zLif{laGbx' ¿sf] gfdnfO{ lardf / 5]psf laGbx' ¿sf] gfdnfO{ 5]pdf /fv]/
n]lvG5 .

dflysf] sf]0fsf] zLif{laGb' B nfO{ lardf /fvL gfd n]Vbf,
A ABC jf CBA

sf]0fnfO{ ;ª\s]t ko| f]u u/L n]lvG5 .
B C ABC nfO{ sf]0f ABC eg]/ kl9G5 .

lrqdf b]vfP h;/L ;an} ] cfkmg\ f] xft p7fpgx' f];\ M

xftnfO{ vD' RofpFbf ag]sf] cfs[lt klg sf]0f xf] .

xftsf] kfv'/f / gf8LnfO{ eh' f / s'lxgfnfO{
zLif{laGb' dfGg ;lsG5 .

AC e'hf AB / eh' f BC zLif{laGb' B df ldn]/
ABC ag]sf] 5 .

B

d/] f] ul0ft, sIff # 75

e'hf, zLifl{ aGb' / sf0] fsf] gfd n]Vg'xf];\ M Z

BX

C D Y /

e'hfx¿ M / eh' fx¿ M
zLif{laGb' M zLif{laGb' M
sf]0f M sf]0f M

E X

F G Y Z

e'hfx¿ M / eh' fx¿ M /

zLif{laGb' M zLif{laGb' M

sf]0f M J sf]0f M

D

G HE F

e'hfx¿ M / e'hfx¿ M /
zLif{laGb' M
sf]0f M zLif{laGb' M

76 d/] f] ul0ft, sIff # sf]0f M

bO' {cf6] f l;Gsfx¿nfO{ eO' Fdf hf8] /] /fVgx' f;] \ . Pp6fnfO{ l:y/ /fvL
csfn{ fO{ tn b]vfP h;/L 3'dfpgx' f];\ . o;/L 3d' fpFbf ag]sf]
km6fOdf cfPsf] cGt/ cjnfs] g ug'{xf];\ M

km6fO a9\b} hfg' eg]sf]
sf]0fsf] gfk klg a9\g' /x]5 .
lhof]af]8{ / /a/AofG8sf] k|of]u u/L ljleGg gfksf sf0] fx¿
agfpg'xf;] \ . tL sf0] fx¿ cjnfs] g u/L ;fgf tyf 7'nf sf]0f
5'6\ofpgx' f;] \ / sIffdf k|:t't ug'x{ f];\ M

d/] f] ul0ft, sIff # 77

tn lbOPh:t} 7n' f] sf0] fdf /ª eg'{xf;] \ M

!= S
P

QR U
T

@= A D

B C EF
J
#=

G

H IK L

$= M P
O QR
N

78 d/] f] ul0ft, sIff #

?n/ ko| f]u u/L laGbx' ¿ hf]8L sf0] f /rgf ug{'xf;] \ M

AA

B CB C

s'g sf]0f 7'nf] aGof] <
s'g sf]0f ;fgf] aGof] <
tn lbOPcg';f/ sf0] fx¿ agfpg'xf;] \ M

ABC DEF

d]/f] ul0ft, sIff # 79

;dsf0] f (Right angle)

tkfOs‘{ f] Hofldlt afs; (Instrument box) leq ePsf tn lrqdf
bv] fOPsf h:tf hf8] L pks/0f lngx' f;] \ / 5nkmn ugx{' f;] \ M

-s_ oL b'j} pks/0f lqeh' cfsf/sf 5g\ . oL pks/0fnfO{ ;]6 :Sjfo/
elgG5 .

-v_ oL bj' } pks/0fsf s'gfx¿nfO{ j|md};Fu vK6\ofpFb} hfg] xf] eg] Pp6f
sg' fdf ag]sf sf]0fx¿ a/fa/ xg' cfpF5g\ .

-u_ tL b'O{ a/fa/ gfksf sf]0fx¿sf] cfsf/ 5, o;sf] sg' fdf
ag]sf] sf]0fnfO{ ;dsf]0f elgG5 .

-3_ ;dsf]0f hgfpg] sf]0fnfO{ ;ª\s]tdf n]lvG5 .
-ª_ ;]6 :Sjfo/sf] ko| f]uaf6 s'g} sf]0f ;dsf]0f eP gePsf] hfFr ug{

;lsG5 .

-r_ tkfO{“sf] kf7\ok:' ts / sfkLsf kfgfsf kT| o]s s'gfdf ;dsf]0f cfs[lt
ag]sf] xG' 5 .

tkfOs{‘ f] j/k/ ePsf j:tx' ¿df sxfF sxfF ;dsf]0f cfs[lt ag]sf]

5 vfh] L u/L n]Vg'xf]; M

80 d]/f] ul0ft, sIff #

sfuh k6\ofO{ ;dsf]0f gfKg] pks/0f agfpg'xf;] \ M

;dsf0] f gfKg] pks/0f ko| f]u u/L ;dsf]0f kQf nufpg'xf;] \ .
s'g} sg' } sf0] fx¿ ;dsf]0feGbf ;fgf jf 7'nf klg 5g\ ls < tn
lbOPsf sf0] fx¿ s:tf s:tf 5g,\ t'ngf ug'{xf];\ M

u8] fu'8L ko| fu] u/L km/s km/s vfnL sfuhsf] kfgfdf ;dsf]0f,
;dsf]0feGbf 7n' f] / ;dsf0] feGbf ;fgf] sf]0f agfpg'xf];\ . o;/L
sf]0fx¿ agfpFbf s;/L agfpge' of] . ;d"xdf 5nkmn u/L
sIffsf7] fdf k:| tt' ugx{' f];\ M

d]/f] ul0ft, sIff # 81

kf7 !! HofldtLo cfsf/x¿

lqeh'

5nkmn ugx'{ f;] \ M

;8ssf lardf s:tf] oxfF sltcf]6f
cfs[lt ag]sf] 5 < s'gfx¿ 5g\ <

;8slardf ag]sf] cfs[ltdf
sltcf]6f l;wf /]vfx¿ 5g\ <

82 d/] f] ul0ft, sIff #

5nkmn ug{x' f];\ M
tLgcf]6f vDafdf nu' f ;'sfpg] 8f]/L 6fFluPsf] 5 . Pp6f nf]vs]{ A :yfgaf6
B x'Fb} C ;Dd ku' ]/ km]l/ A ;Dd cfP/ Ps kmGsf] nufof] .

A

CB

nf]vs]{ lxF8]sf] 8f]/Lsf] af6f] lqe'h xf] .
s:tf] cfs[ltsf] 5 <

?n/sf] ko| fu] u/L lbOPsf laGb'x¿nfO{ hf]8/] aGb cfsl[ t
agfpg'xf;] \ M

A PR

BC Q

d/] f] ul0ft, sIff # 83

5nkmn ugx{' f];\ M

A B
C
;Fu}sf] lqeh' nfO{ lqe'h ABC jf
∆ ABC elgG5 .

∆ ABC sf e'hfx¿ sg' sg' xg' \ <
AB, BC / CA xg' \ .
l7s eGg'eof] .

clg zLif{laGbx' ¿ sg' s'g xg' \ <
zLif{laGb'x¿ A, B / C xg' \ .
l7s eGg'eof] .

84 d]/f] ul0ft, sIff #

e'hfx¿, zLifl{ aGb'x¿ / lqeh' sf] gfd n]Vgx' f];\ M

P e'hfx¿M PQ, QR / RP
zLif{laGb'x¿M P, Q / R

Q
R ∆ PQR

G eh' fx¿M , ,
H zLif{laGb'x¿M , ,

K

∆

O e'hfx¿M , ,
zLif{laGb'x¿M , ,

M N∆

S e'hfx¿M , ,

zLif{laGb'x¿M , ,

TU
∆

d]/f] ul0ft, sIff # 85

rt'eh'{

5nkmn ugx'{ f;] \ M

lbOPsf] lrqdf 3/ / cfFugdf s:tf cfs[ltx¿ 5g\ <

kT| o]s cfs[ltdf $
sltcf]6f eh' fx¿ 5g\ < $

k|To]s cfs[ltdf
sltcf]6f s'gfx¿ 5g\ <

rf/ sg' f / rf/ e'hf rt'e{'h
ePsf] aGb cfs[ltnfO{
s] elgG5 <

rf/cf]6f l;wf /]vfx¿ ldn]/ ag]sf] aGb cfs[ltnfO{ rte' {'h elgG5 .

lbOPsf] rt'e{h' nfO{ rt'e'{h ABCD elgG5 . D C
laGb'x¿ A,B,C / D nfO{ rt'e{'h ABCD sf A B
zLif{laGb'x¿ elgG5 .

/]vfv08x¿ AB, BC, CD / DA nfO{
rt'e{'h ABCD sf eh' fx¿ elgG5 .

86 d/] f] ul0ft, sIff #

rt'e'h{

l;wf lsgf/f ePsf j:t' k|ofu] u/L rt'e{'h lvRgx' f];\ M

lbOPsf rf/cf6] f laGb'x¿nfO{ ?n/sf ;xfotfn] jm| d;} Fu
hf]8g\ 'xf;] \ / aGb cfs[lt agfpgx' f];\ M

A DP S

B CQ R

d]/f] ul0ft, sIff # 87

HofldtLo cfsl[ t

;dsf0] f gfKg] pks/0f k|of]u u/L tn lbOPsf rte' h{' x¿dWo]
rf/cf]6} sf]0f ;dsf0] f ePsf rt'e{h' x¿ klxrfg u/L /ª egx{' f;] \ M

?n/ k|of]u u/L rf/cf6] } eh' fx¿sf] gfk a/fa/ ePsf rt'{e'h
s'g s'g x'g,\ kQf nufpgx' f];\ / /ª eg{x' f];\ M

88 d]/f] ul0ft, sIff #

ju{ / cfot

tn elgPsf ljm| ofsnfkx¿ ugx'{ f;] \ M

Pp6f cfotfsf/ sfuhsf] Pp6f sg' f A df Pp6f cfn¹ fn] lyr]/
kfgf lngx' f];\ . csf]{ xftn] csf]{ sg' f B nfO{ prfn]/
lsgf/f AD df vK6\ofpgx' f];\ M
AD
A BD

BC C

afFsL /x]sf] efunfO{ vlK6P/ afFsL ePsf]
km'sfP/ x]g{'xf];\ . efunfO{ sf6]/ jf
Roft]/ 5'6\ofpgx' f];\ M

o;df ;a} e'hfsf] nDafO gfKg'xf];\ / s:tf] rte' {'h aGof],
5nkmn ug'{xf];\ .

tkfO{“n] ;?' df lnPsf] sfkLsf] kfgfsf] cfsf/
cfotfsf/ lyof] eg] clGtddf afFsL /x]sf]
efusf] cfsf/ juf{sf/ ag]sf] 5 .

d]/f] ul0ft, sIff # 89

tn elgPsf ljm| ofsnfkx¿ ugx'{ f];\ M

sfkLsf] Pp6f kfgf lng'xf];\ M bfofF / afofFsf lsgf/fx¿
cfk;df vlK6g] u/L laraf6
k6\ofpg'xf];\ M

o;nfO{ km'sfpgx' f];\ / tn /
dflysf lsgf/fx¿ cfk;df
vlK6g] u/L k6\ofpgx' f];\ M

cfotsf bfofF / afofFsf lsgf/fx¿ cfk;df a/fa/ x'Fbf /x]5g\ .
To;} u/L tn / dflysf lsgf/fx¿ klg cfk;df a/fa/ x'Fbf /x]5g\ .

90 d]/f] ul0ft, sIff #

tn elgPsf ljm| ofsnfkx¿ ug{'xf;] \ M
Pp6f juf{sf/ sfuhsf] kfgf lng'xf];\ M

lrqdf b]vfOP h:t} ljk/Lt sg' fx¿ cfk;df vlK6g] u/L laraf6
k6\ofpg'xf];\ M

km]l/ lrqdf b]vfOP h:t} ljk/Lt s'gfx¿ cfk;df vlK6g] u/L laraf6
k6\ofpg'xf];\ M

ju{sf ;a} lsgf/fx¿ cfk;df a/fa/ xF'bf /x]5g\ .

d/] f] ul0ft, sIff # 91

5nkmn ug{x' f;] \ M cfotsf ;fd'Gg]sf
e'hf / rf/cf]6} sf]0f
ju{sf rf/cf]6}
eh' f / rf/cf]6} a/fa/ xG' 5g\ .

sf]0f a/fa/
xG' 5g\ .

of] ju{ xf] . of] cfot xf] .

tnsf jud{ f 9frF f / cfotdf 9fFrf agfpg'xf;] \ M

92 d/] f] ul0ft, sIff #

cfotfsf/ ;tx ePsf j:t'df l7s (√) / jufs{ f/ ;tx ePsf
j:td' f ufn] f] 3/] f ( ) nufpgx' f;] \ M

d]/f] ul0ft, sIff # 93

?n/sf] ko| fu] u/L yfK] nfx¿ hf8] ]/ lqe'h, ju{ / cfot cfsf/sf]
Ps Pscf6] f cfs[lt agfpg'xf];\ M

Pp6f vn] v]nf‘} M
a/fa/ gfksf !) cf]6f l;Gsfx¿ jf l;;fsndx¿
jf ;nfOsf sfF6Lx¿ jf o:t} cGo s'g} j:tx' ¿
lnP/ oxfF lbPsf] h:t} cfs[lt agfpgx' f];\ M

tl/sf ! M sg' } bO' {cf]6f sfF6Lx¿ x6fP/ b'O{cf]6f ju{x¿ agfpgx' f];\ .
tl/sf @ M sg' } Pp6f sfF6L x6fP/ Pp6f cfot / Pp6f ju{ agfpgx' f];\ .

94 d]/f] ul0ft, sIff #