NCERT पाठ्यपुस्तक पाठ 11 - रचनाएँ, कक्षा 10, गणित Class 10 Notes | EduRev

गणित कक्षा 10

Class 10 : NCERT पाठ्यपुस्तक पाठ 11 - रचनाएँ, कक्षा 10, गणित Class 10 Notes | EduRev

 Page 1


238 xf.kr
11
11.1 Hkwfedk
d{kk IX  esa] vkius ,d iVjh rFkk ijdkj dk iz;ksx djosQ oqQN jpuk,¡ dh Fkh] tSls fdlh
dks.k dks lef}Hkkftr djuk] fdlh js[kk[kaM dk yac lef}Hkktd [khapuk] oqQN f=kHkqtksa dh
jpuk,¡ djuk bR;kfn rFkk mudk vkSfpR; Hkh fn;k FkkA bl vè;k; esa] ge fiNyh jpukvksa
osQ Kku dk mi;ksx djrs gq,] oqQN vkSj jpukvksa dk vè;;u djsaxsA ;s jpuk,¡ D;ksa gks
tkrh gSa] buls lacaf/r oqQN xf.krh; O;k[;k Hkh vkidks nsuh gksxhA
11.2 js[kk[kaM dk foHkktu
eku yhft, fd ,d js[kk[kaM fn;k gS vkSj vkidks mls ,d fn, x, vuqikr] ekuk 3 : 2
esa foHkkftr djuk gSA vki bldh yackbZ eki dj rFkk fn, x, vuqikr osQ vuqlkj ,d
¯cnq fpfÉr dj ldrs gSaA ijarq ;fn vkiosQ ikl bls lgh&lgh ekius dh dksbZ fof/ u
gks] rks vki bl ¯cnq dks oSQls izkIr djsaxs\ bl izdkj osQ ¯cnq dks izkIr djus osQ fy,]
ge fuEufyf[kr nks fof/;k¡ ns jgs gSa%
jpuk 11.1 : ,d js[kk[kaM dks fn, gq, vuqikr esa foHkkftr djukA
,d js[kk[kaM AB fn;k gS] ge bldks m : n osQ vuqikr esa foHkkftr djuk pkgrs gSaA
izfØ;k dks le>us esa lgk;rk djus osQ fy,] ge m = 3 vkSj  n = 2  ysaxsA
jpuk osQ pj.k%
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. AX ij 5 (= m + n) ¯cnq  A
1
, A
2
, A
3
, A
4
 vkSj A
5
 bl izdkj vafdr dhft, fd
AA
1
 = A
1
A
2
 = A
2
A
3
 = A
3
A
4
 = A
4
A
5
 gksA
3. BA
5  
dks feykb,A
jpuk,¡
Page 2


238 xf.kr
11
11.1 Hkwfedk
d{kk IX  esa] vkius ,d iVjh rFkk ijdkj dk iz;ksx djosQ oqQN jpuk,¡ dh Fkh] tSls fdlh
dks.k dks lef}Hkkftr djuk] fdlh js[kk[kaM dk yac lef}Hkktd [khapuk] oqQN f=kHkqtksa dh
jpuk,¡ djuk bR;kfn rFkk mudk vkSfpR; Hkh fn;k FkkA bl vè;k; esa] ge fiNyh jpukvksa
osQ Kku dk mi;ksx djrs gq,] oqQN vkSj jpukvksa dk vè;;u djsaxsA ;s jpuk,¡ D;ksa gks
tkrh gSa] buls lacaf/r oqQN xf.krh; O;k[;k Hkh vkidks nsuh gksxhA
11.2 js[kk[kaM dk foHkktu
eku yhft, fd ,d js[kk[kaM fn;k gS vkSj vkidks mls ,d fn, x, vuqikr] ekuk 3 : 2
esa foHkkftr djuk gSA vki bldh yackbZ eki dj rFkk fn, x, vuqikr osQ vuqlkj ,d
¯cnq fpfÉr dj ldrs gSaA ijarq ;fn vkiosQ ikl bls lgh&lgh ekius dh dksbZ fof/ u
gks] rks vki bl ¯cnq dks oSQls izkIr djsaxs\ bl izdkj osQ ¯cnq dks izkIr djus osQ fy,]
ge fuEufyf[kr nks fof/;k¡ ns jgs gSa%
jpuk 11.1 : ,d js[kk[kaM dks fn, gq, vuqikr esa foHkkftr djukA
,d js[kk[kaM AB fn;k gS] ge bldks m : n osQ vuqikr esa foHkkftr djuk pkgrs gSaA
izfØ;k dks le>us esa lgk;rk djus osQ fy,] ge m = 3 vkSj  n = 2  ysaxsA
jpuk osQ pj.k%
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. AX ij 5 (= m + n) ¯cnq  A
1
, A
2
, A
3
, A
4
 vkSj A
5
 bl izdkj vafdr dhft, fd
AA
1
 = A
1
A
2
 = A
2
A
3
 = A
3
A
4
 = A
4
A
5
 gksA
3. BA
5  
dks feykb,A
jpuk,¡
jpuk,¡ 239
4. ¯cnq A
3
 (m = 3) ls gksdj tkus okyh  A
5
B osQ
lekarj ,d js[kk (A
3
 ij ? AA
5
B osQ cjkcj
dks.k cukdj) AB dks ,d ¯cnq C ij izfrPNsn
djrh gqbZ [khafp, (nsf[k, vko`Qfr 11-1)A
rc] AC : CB = 3 : 2 gSA
vkb, ns[ksa fd ;g fof/ oSQls gesa vHkh"V foHkktu nsrh gSA
D;ksafd A
3
C, A
5
B  osQ lekarj gS]
vr%
3
3 5
AA
AA
 = 
AC
CB
(vk/kjHkwr lekuqikfrdrk izes; }kjk)
jpuk ls] 
3
3 5
AA 3 AC 3
A A 2 CB 2
= = gA S vr% gSA
blls ;g fu"d"kZ fudyrk gS fd fcanq C, AB dks 3 : 2
vuqikr esa foHkkftr djrk gSA
oSdfYid fof/
jpuk osQ pj.k :
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. ?BAX osQ cjkcj ?ABY cukdj  AX  osQ lekarj ,d fdj.k BY [khafp,A
3. AX ij ¯cnq A
1
, A
2
, A
3
 (m = 3) vkSj BY ij fcanq B
1
, B
2
 (n = 2) bl izdkj vafdr
dhft, fd AA
1
 = A
1
A
2
 = A
2
A
3
 = BB
1
 = B
1
B
2 
gksA
4. A
3
B
2  
dks feykb,A ekuk ;g AB dks ¯cnq C ij izfrPNsn djrh gS (nsf[k, vko`Qfr 11-2)A
rc] AC : CB = 3 : 2 gSA
vkb, ns[kas fd bl fof/ ls gesa vHkh"V jpuk fdl izdkj izkIr gksrk gS\
;gk¡ ? AA
3
C ~ ? BB
2
C (D;ksa\)
rc
3
2
AA AC
BB BC
=
ijarq jpuk }kjk 
3
2
AA 3
BB 2
=
 gSA vr %] 
AC 3
BC 2
=
vko`Qfr 11.2
vko`Qfr 11.1
Page 3


238 xf.kr
11
11.1 Hkwfedk
d{kk IX  esa] vkius ,d iVjh rFkk ijdkj dk iz;ksx djosQ oqQN jpuk,¡ dh Fkh] tSls fdlh
dks.k dks lef}Hkkftr djuk] fdlh js[kk[kaM dk yac lef}Hkktd [khapuk] oqQN f=kHkqtksa dh
jpuk,¡ djuk bR;kfn rFkk mudk vkSfpR; Hkh fn;k FkkA bl vè;k; esa] ge fiNyh jpukvksa
osQ Kku dk mi;ksx djrs gq,] oqQN vkSj jpukvksa dk vè;;u djsaxsA ;s jpuk,¡ D;ksa gks
tkrh gSa] buls lacaf/r oqQN xf.krh; O;k[;k Hkh vkidks nsuh gksxhA
11.2 js[kk[kaM dk foHkktu
eku yhft, fd ,d js[kk[kaM fn;k gS vkSj vkidks mls ,d fn, x, vuqikr] ekuk 3 : 2
esa foHkkftr djuk gSA vki bldh yackbZ eki dj rFkk fn, x, vuqikr osQ vuqlkj ,d
¯cnq fpfÉr dj ldrs gSaA ijarq ;fn vkiosQ ikl bls lgh&lgh ekius dh dksbZ fof/ u
gks] rks vki bl ¯cnq dks oSQls izkIr djsaxs\ bl izdkj osQ ¯cnq dks izkIr djus osQ fy,]
ge fuEufyf[kr nks fof/;k¡ ns jgs gSa%
jpuk 11.1 : ,d js[kk[kaM dks fn, gq, vuqikr esa foHkkftr djukA
,d js[kk[kaM AB fn;k gS] ge bldks m : n osQ vuqikr esa foHkkftr djuk pkgrs gSaA
izfØ;k dks le>us esa lgk;rk djus osQ fy,] ge m = 3 vkSj  n = 2  ysaxsA
jpuk osQ pj.k%
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. AX ij 5 (= m + n) ¯cnq  A
1
, A
2
, A
3
, A
4
 vkSj A
5
 bl izdkj vafdr dhft, fd
AA
1
 = A
1
A
2
 = A
2
A
3
 = A
3
A
4
 = A
4
A
5
 gksA
3. BA
5  
dks feykb,A
jpuk,¡
jpuk,¡ 239
4. ¯cnq A
3
 (m = 3) ls gksdj tkus okyh  A
5
B osQ
lekarj ,d js[kk (A
3
 ij ? AA
5
B osQ cjkcj
dks.k cukdj) AB dks ,d ¯cnq C ij izfrPNsn
djrh gqbZ [khafp, (nsf[k, vko`Qfr 11-1)A
rc] AC : CB = 3 : 2 gSA
vkb, ns[ksa fd ;g fof/ oSQls gesa vHkh"V foHkktu nsrh gSA
D;ksafd A
3
C, A
5
B  osQ lekarj gS]
vr%
3
3 5
AA
AA
 = 
AC
CB
(vk/kjHkwr lekuqikfrdrk izes; }kjk)
jpuk ls] 
3
3 5
AA 3 AC 3
A A 2 CB 2
= = gA S vr% gSA
blls ;g fu"d"kZ fudyrk gS fd fcanq C, AB dks 3 : 2
vuqikr esa foHkkftr djrk gSA
oSdfYid fof/
jpuk osQ pj.k :
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. ?BAX osQ cjkcj ?ABY cukdj  AX  osQ lekarj ,d fdj.k BY [khafp,A
3. AX ij ¯cnq A
1
, A
2
, A
3
 (m = 3) vkSj BY ij fcanq B
1
, B
2
 (n = 2) bl izdkj vafdr
dhft, fd AA
1
 = A
1
A
2
 = A
2
A
3
 = BB
1
 = B
1
B
2 
gksA
4. A
3
B
2  
dks feykb,A ekuk ;g AB dks ¯cnq C ij izfrPNsn djrh gS (nsf[k, vko`Qfr 11-2)A
rc] AC : CB = 3 : 2 gSA
vkb, ns[kas fd bl fof/ ls gesa vHkh"V jpuk fdl izdkj izkIr gksrk gS\
;gk¡ ? AA
3
C ~ ? BB
2
C (D;ksa\)
rc
3
2
AA AC
BB BC
=
ijarq jpuk }kjk 
3
2
AA 3
BB 2
=
 gSA vr %] 
AC 3
BC 2
=
vko`Qfr 11.2
vko`Qfr 11.1
240 xf.kr
vko`Qfr 11.3
okLro esa bu fof/;ksa }kjk fn;s x;s js[kk[kaM dks fdlh Hkh vuqikr esa foHkkftr
fd;k tk ldrk gSA
vc ge Åij nh xbZ jpuk dks ,d fn, x, f=kHkqt osQ le:i ,d vU; f=kHkqt
dh jpuk djus esa mi;ksx djsaxs ftldh Hkqtkvksa vkSj fn, x, f=kHkqt dh laxr
Hkqtkvksa esa ,d vuqikr fn;k gqvk gksA
jpuk 11.2 : ,d fn, x, LosQy xq.kd osQ vuqlkj fn, x, f=kHkqt osQ le:i ,d
f=kHkqt dh jpuk djukA
bl jpuk dh nks fLFkfr;k¡ gSaA ,d esa] ftl f=kHkqt dh jpuk djuh gS] og fn,
x, f=kHkqt ls NksVk gks rFkk nwljh esa og cM+k gksA ;gk¡ LosQy xq.kd dk vFkZ jpuk djus
okys f=kHkqt dh Hkqtkvksa rFkk fn, gq, f=kHkqt dh laxr Hkqtkvksa osQ vuqikr ls gSA
(vè;k; 6 Hkh nsf[k,)A bu jpukvksa dks le>us osQ fy, vkb, fuEu mnkgj.k ysaA
;gh fof/ O;kid fLFkfr esa Hkh ykxw gksxhA
mnkgj.k 1 : ,d fn, x, f=kHkqt ABC osQ le:i ,d f=kHkqt dh jpuk dhft,]
ftldh Hkqtk,¡ fn, x, f=kHkqt dh laxr Hkqtkvksa dh 
3
4
 gksa (vFkkZr~ LosQy xq.kd 
3
4
 gS)A
gy : ,d f=kHkqt ABC fn;k gSA gesa ,d vU; f=kHkqt dh jpuk djuh gS] ftldh
Hkqtk,¡ f=kHkqt ABC dh laxr Hkqtkvksa dh 
3
4
 gksaA
jpuk osQ pj.k%
1. BC ls 'kh"kZ A  dh nwljh vksj U;wudks.k cukrh gqbZ ,d fdj.k BX  [khafp,A
2. BX ij 4 fcanq (
3
4
 esa 3 vkSj 4 esa ls cM+h la[;k) B
1
, B
2
, B
3
 vkSj B
4
, bl izdkj
vafdr dhft, fd BB
1
 = B
1
B
2
 = B
2
B
3
 = B
3
B
4 
gksA
3. B
4
C feykb, vkSj B
3
 (rhljs ¯cnq] ;gk¡
3
4
 esa 3 vkSj 4 esa ls 3 NksVh gS) ls
gksdj tkus okyh B
4
C osQ lekarj ,d js[kk
BC dks C'  ij izfrPNsn djrh gqbZ [khafp,A
4. C' ls gksdj tkus okyh CA  osQ lekarj
,d js[kk BA dks A' ij izfrPNsn djrh
gqbZ [khafp, (nsf[k, vko`Qfr 11-3)A
rc] ? A'BC' vHkh"V f=kHkqt gSA
Page 4


238 xf.kr
11
11.1 Hkwfedk
d{kk IX  esa] vkius ,d iVjh rFkk ijdkj dk iz;ksx djosQ oqQN jpuk,¡ dh Fkh] tSls fdlh
dks.k dks lef}Hkkftr djuk] fdlh js[kk[kaM dk yac lef}Hkktd [khapuk] oqQN f=kHkqtksa dh
jpuk,¡ djuk bR;kfn rFkk mudk vkSfpR; Hkh fn;k FkkA bl vè;k; esa] ge fiNyh jpukvksa
osQ Kku dk mi;ksx djrs gq,] oqQN vkSj jpukvksa dk vè;;u djsaxsA ;s jpuk,¡ D;ksa gks
tkrh gSa] buls lacaf/r oqQN xf.krh; O;k[;k Hkh vkidks nsuh gksxhA
11.2 js[kk[kaM dk foHkktu
eku yhft, fd ,d js[kk[kaM fn;k gS vkSj vkidks mls ,d fn, x, vuqikr] ekuk 3 : 2
esa foHkkftr djuk gSA vki bldh yackbZ eki dj rFkk fn, x, vuqikr osQ vuqlkj ,d
¯cnq fpfÉr dj ldrs gSaA ijarq ;fn vkiosQ ikl bls lgh&lgh ekius dh dksbZ fof/ u
gks] rks vki bl ¯cnq dks oSQls izkIr djsaxs\ bl izdkj osQ ¯cnq dks izkIr djus osQ fy,]
ge fuEufyf[kr nks fof/;k¡ ns jgs gSa%
jpuk 11.1 : ,d js[kk[kaM dks fn, gq, vuqikr esa foHkkftr djukA
,d js[kk[kaM AB fn;k gS] ge bldks m : n osQ vuqikr esa foHkkftr djuk pkgrs gSaA
izfØ;k dks le>us esa lgk;rk djus osQ fy,] ge m = 3 vkSj  n = 2  ysaxsA
jpuk osQ pj.k%
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. AX ij 5 (= m + n) ¯cnq  A
1
, A
2
, A
3
, A
4
 vkSj A
5
 bl izdkj vafdr dhft, fd
AA
1
 = A
1
A
2
 = A
2
A
3
 = A
3
A
4
 = A
4
A
5
 gksA
3. BA
5  
dks feykb,A
jpuk,¡
jpuk,¡ 239
4. ¯cnq A
3
 (m = 3) ls gksdj tkus okyh  A
5
B osQ
lekarj ,d js[kk (A
3
 ij ? AA
5
B osQ cjkcj
dks.k cukdj) AB dks ,d ¯cnq C ij izfrPNsn
djrh gqbZ [khafp, (nsf[k, vko`Qfr 11-1)A
rc] AC : CB = 3 : 2 gSA
vkb, ns[ksa fd ;g fof/ oSQls gesa vHkh"V foHkktu nsrh gSA
D;ksafd A
3
C, A
5
B  osQ lekarj gS]
vr%
3
3 5
AA
AA
 = 
AC
CB
(vk/kjHkwr lekuqikfrdrk izes; }kjk)
jpuk ls] 
3
3 5
AA 3 AC 3
A A 2 CB 2
= = gA S vr% gSA
blls ;g fu"d"kZ fudyrk gS fd fcanq C, AB dks 3 : 2
vuqikr esa foHkkftr djrk gSA
oSdfYid fof/
jpuk osQ pj.k :
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. ?BAX osQ cjkcj ?ABY cukdj  AX  osQ lekarj ,d fdj.k BY [khafp,A
3. AX ij ¯cnq A
1
, A
2
, A
3
 (m = 3) vkSj BY ij fcanq B
1
, B
2
 (n = 2) bl izdkj vafdr
dhft, fd AA
1
 = A
1
A
2
 = A
2
A
3
 = BB
1
 = B
1
B
2 
gksA
4. A
3
B
2  
dks feykb,A ekuk ;g AB dks ¯cnq C ij izfrPNsn djrh gS (nsf[k, vko`Qfr 11-2)A
rc] AC : CB = 3 : 2 gSA
vkb, ns[kas fd bl fof/ ls gesa vHkh"V jpuk fdl izdkj izkIr gksrk gS\
;gk¡ ? AA
3
C ~ ? BB
2
C (D;ksa\)
rc
3
2
AA AC
BB BC
=
ijarq jpuk }kjk 
3
2
AA 3
BB 2
=
 gSA vr %] 
AC 3
BC 2
=
vko`Qfr 11.2
vko`Qfr 11.1
240 xf.kr
vko`Qfr 11.3
okLro esa bu fof/;ksa }kjk fn;s x;s js[kk[kaM dks fdlh Hkh vuqikr esa foHkkftr
fd;k tk ldrk gSA
vc ge Åij nh xbZ jpuk dks ,d fn, x, f=kHkqt osQ le:i ,d vU; f=kHkqt
dh jpuk djus esa mi;ksx djsaxs ftldh Hkqtkvksa vkSj fn, x, f=kHkqt dh laxr
Hkqtkvksa esa ,d vuqikr fn;k gqvk gksA
jpuk 11.2 : ,d fn, x, LosQy xq.kd osQ vuqlkj fn, x, f=kHkqt osQ le:i ,d
f=kHkqt dh jpuk djukA
bl jpuk dh nks fLFkfr;k¡ gSaA ,d esa] ftl f=kHkqt dh jpuk djuh gS] og fn,
x, f=kHkqt ls NksVk gks rFkk nwljh esa og cM+k gksA ;gk¡ LosQy xq.kd dk vFkZ jpuk djus
okys f=kHkqt dh Hkqtkvksa rFkk fn, gq, f=kHkqt dh laxr Hkqtkvksa osQ vuqikr ls gSA
(vè;k; 6 Hkh nsf[k,)A bu jpukvksa dks le>us osQ fy, vkb, fuEu mnkgj.k ysaA
;gh fof/ O;kid fLFkfr esa Hkh ykxw gksxhA
mnkgj.k 1 : ,d fn, x, f=kHkqt ABC osQ le:i ,d f=kHkqt dh jpuk dhft,]
ftldh Hkqtk,¡ fn, x, f=kHkqt dh laxr Hkqtkvksa dh 
3
4
 gksa (vFkkZr~ LosQy xq.kd 
3
4
 gS)A
gy : ,d f=kHkqt ABC fn;k gSA gesa ,d vU; f=kHkqt dh jpuk djuh gS] ftldh
Hkqtk,¡ f=kHkqt ABC dh laxr Hkqtkvksa dh 
3
4
 gksaA
jpuk osQ pj.k%
1. BC ls 'kh"kZ A  dh nwljh vksj U;wudks.k cukrh gqbZ ,d fdj.k BX  [khafp,A
2. BX ij 4 fcanq (
3
4
 esa 3 vkSj 4 esa ls cM+h la[;k) B
1
, B
2
, B
3
 vkSj B
4
, bl izdkj
vafdr dhft, fd BB
1
 = B
1
B
2
 = B
2
B
3
 = B
3
B
4 
gksA
3. B
4
C feykb, vkSj B
3
 (rhljs ¯cnq] ;gk¡
3
4
 esa 3 vkSj 4 esa ls 3 NksVh gS) ls
gksdj tkus okyh B
4
C osQ lekarj ,d js[kk
BC dks C'  ij izfrPNsn djrh gqbZ [khafp,A
4. C' ls gksdj tkus okyh CA  osQ lekarj
,d js[kk BA dks A' ij izfrPNsn djrh
gqbZ [khafp, (nsf[k, vko`Qfr 11-3)A
rc] ? A'BC' vHkh"V f=kHkqt gSA
jpuk,¡ 241
vkb, ns[ksa fd bl jpuk ls oSQls vHkh"V f=kHkqt izkIr gks tkrk gSA
jpuk 11-1 ls]  
BC 3
CC 1
'
=
'
blfy,] 
BC BC + C C C C 1 4
1 1
BC BC BC 3 3
' ' '
= = + = + =
' ' '
, vFkkZr~ 
BC
BC
'
 = 
3
4
 gSA
lkFk gh] C'A', CA osQ lekarj gSA blfy,  ? A'BC' ~ ? ABC (D;ksa\)
vr%]
A B A C BC 3
AB AC BC 4
' ' ' '
= = =
mnkgj.k 2 : ,d fn, x, f=kHkqt  ABC osQ le:i ,d f=kHkqt dh jpuk dhft,]
ftldh Hkqtk,¡ f=kHkqt ABC dh laxr Hkqtkvksa dh 
5
3
 gksa (vFkkZr~ LosQy xq.kd 
5
3
 gS)A
gy : ,d f=kHkqt ABC fn;k x;k gSA gesa ,d f=kHkqt dh jpuk djuh gS] ftldh Hkqtk,¡
? ABC dh laxr Hkqtkvksa dh 
5
3
 gksaA
jpuk osQ pj.k%
1. BC ls 'kh"kZ A osQ nwljh vksj U;wudks.k cukrh gqbZ ,d fdj.k BX [khafp,A
2. 5 (
5
3
 esa 5 vkSj 3 esa ls cM+h la[;k) ¯cnq B
1
, B
2
, B
3
, B
4
 vkSj B
5
, BX ij bl izdkj
vafdr dhft, fd BB
1
 = B
1
B
2
 = B
2
B
3
 = B
3
B
4
 = B
4
B
5 
gksA
3. B
3
 (rhljk ¯cnq] 
5
3
 esa 5 vkSj 3 esa ls NksVh la[;k) dks C ls feykb, vkSj B
5
 ls
gksdj tkus okyh  B
3
C osQ lekarj ,d js[kk] c<+k, x, js[kk[kaM BC dks C' ij
izfrPNsn djrh gqbZ [khafp,A
4. C' ls gksdj tkus okyh CA osQ lekarj ,d
js[kk] c<+kus ij js[kk[kaM BA dks A' ij izfrPNsn
djrh gqbZ [khafp, (nsf[k, vko`Qfr 11-4)A
rc] A'BC' vHkh"V f=kHkqt gSA
jpuk osQ vkSfpR; fl¼ djus osQ fy,]
è;ku nhft,  ? ABC ~ ? A'BC' (D;ksa\)
blfy, 
AB AC BC
AB AC BC
= =
' ' ' '
 gSA
vko`Qfr 11.4
Page 5


238 xf.kr
11
11.1 Hkwfedk
d{kk IX  esa] vkius ,d iVjh rFkk ijdkj dk iz;ksx djosQ oqQN jpuk,¡ dh Fkh] tSls fdlh
dks.k dks lef}Hkkftr djuk] fdlh js[kk[kaM dk yac lef}Hkktd [khapuk] oqQN f=kHkqtksa dh
jpuk,¡ djuk bR;kfn rFkk mudk vkSfpR; Hkh fn;k FkkA bl vè;k; esa] ge fiNyh jpukvksa
osQ Kku dk mi;ksx djrs gq,] oqQN vkSj jpukvksa dk vè;;u djsaxsA ;s jpuk,¡ D;ksa gks
tkrh gSa] buls lacaf/r oqQN xf.krh; O;k[;k Hkh vkidks nsuh gksxhA
11.2 js[kk[kaM dk foHkktu
eku yhft, fd ,d js[kk[kaM fn;k gS vkSj vkidks mls ,d fn, x, vuqikr] ekuk 3 : 2
esa foHkkftr djuk gSA vki bldh yackbZ eki dj rFkk fn, x, vuqikr osQ vuqlkj ,d
¯cnq fpfÉr dj ldrs gSaA ijarq ;fn vkiosQ ikl bls lgh&lgh ekius dh dksbZ fof/ u
gks] rks vki bl ¯cnq dks oSQls izkIr djsaxs\ bl izdkj osQ ¯cnq dks izkIr djus osQ fy,]
ge fuEufyf[kr nks fof/;k¡ ns jgs gSa%
jpuk 11.1 : ,d js[kk[kaM dks fn, gq, vuqikr esa foHkkftr djukA
,d js[kk[kaM AB fn;k gS] ge bldks m : n osQ vuqikr esa foHkkftr djuk pkgrs gSaA
izfØ;k dks le>us esa lgk;rk djus osQ fy,] ge m = 3 vkSj  n = 2  ysaxsA
jpuk osQ pj.k%
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. AX ij 5 (= m + n) ¯cnq  A
1
, A
2
, A
3
, A
4
 vkSj A
5
 bl izdkj vafdr dhft, fd
AA
1
 = A
1
A
2
 = A
2
A
3
 = A
3
A
4
 = A
4
A
5
 gksA
3. BA
5  
dks feykb,A
jpuk,¡
jpuk,¡ 239
4. ¯cnq A
3
 (m = 3) ls gksdj tkus okyh  A
5
B osQ
lekarj ,d js[kk (A
3
 ij ? AA
5
B osQ cjkcj
dks.k cukdj) AB dks ,d ¯cnq C ij izfrPNsn
djrh gqbZ [khafp, (nsf[k, vko`Qfr 11-1)A
rc] AC : CB = 3 : 2 gSA
vkb, ns[ksa fd ;g fof/ oSQls gesa vHkh"V foHkktu nsrh gSA
D;ksafd A
3
C, A
5
B  osQ lekarj gS]
vr%
3
3 5
AA
AA
 = 
AC
CB
(vk/kjHkwr lekuqikfrdrk izes; }kjk)
jpuk ls] 
3
3 5
AA 3 AC 3
A A 2 CB 2
= = gA S vr% gSA
blls ;g fu"d"kZ fudyrk gS fd fcanq C, AB dks 3 : 2
vuqikr esa foHkkftr djrk gSA
oSdfYid fof/
jpuk osQ pj.k :
1. AB ls U;wudks.k cukrh dksbZ fdj.k AX [khafp,A
2. ?BAX osQ cjkcj ?ABY cukdj  AX  osQ lekarj ,d fdj.k BY [khafp,A
3. AX ij ¯cnq A
1
, A
2
, A
3
 (m = 3) vkSj BY ij fcanq B
1
, B
2
 (n = 2) bl izdkj vafdr
dhft, fd AA
1
 = A
1
A
2
 = A
2
A
3
 = BB
1
 = B
1
B
2 
gksA
4. A
3
B
2  
dks feykb,A ekuk ;g AB dks ¯cnq C ij izfrPNsn djrh gS (nsf[k, vko`Qfr 11-2)A
rc] AC : CB = 3 : 2 gSA
vkb, ns[kas fd bl fof/ ls gesa vHkh"V jpuk fdl izdkj izkIr gksrk gS\
;gk¡ ? AA
3
C ~ ? BB
2
C (D;ksa\)
rc
3
2
AA AC
BB BC
=
ijarq jpuk }kjk 
3
2
AA 3
BB 2
=
 gSA vr %] 
AC 3
BC 2
=
vko`Qfr 11.2
vko`Qfr 11.1
240 xf.kr
vko`Qfr 11.3
okLro esa bu fof/;ksa }kjk fn;s x;s js[kk[kaM dks fdlh Hkh vuqikr esa foHkkftr
fd;k tk ldrk gSA
vc ge Åij nh xbZ jpuk dks ,d fn, x, f=kHkqt osQ le:i ,d vU; f=kHkqt
dh jpuk djus esa mi;ksx djsaxs ftldh Hkqtkvksa vkSj fn, x, f=kHkqt dh laxr
Hkqtkvksa esa ,d vuqikr fn;k gqvk gksA
jpuk 11.2 : ,d fn, x, LosQy xq.kd osQ vuqlkj fn, x, f=kHkqt osQ le:i ,d
f=kHkqt dh jpuk djukA
bl jpuk dh nks fLFkfr;k¡ gSaA ,d esa] ftl f=kHkqt dh jpuk djuh gS] og fn,
x, f=kHkqt ls NksVk gks rFkk nwljh esa og cM+k gksA ;gk¡ LosQy xq.kd dk vFkZ jpuk djus
okys f=kHkqt dh Hkqtkvksa rFkk fn, gq, f=kHkqt dh laxr Hkqtkvksa osQ vuqikr ls gSA
(vè;k; 6 Hkh nsf[k,)A bu jpukvksa dks le>us osQ fy, vkb, fuEu mnkgj.k ysaA
;gh fof/ O;kid fLFkfr esa Hkh ykxw gksxhA
mnkgj.k 1 : ,d fn, x, f=kHkqt ABC osQ le:i ,d f=kHkqt dh jpuk dhft,]
ftldh Hkqtk,¡ fn, x, f=kHkqt dh laxr Hkqtkvksa dh 
3
4
 gksa (vFkkZr~ LosQy xq.kd 
3
4
 gS)A
gy : ,d f=kHkqt ABC fn;k gSA gesa ,d vU; f=kHkqt dh jpuk djuh gS] ftldh
Hkqtk,¡ f=kHkqt ABC dh laxr Hkqtkvksa dh 
3
4
 gksaA
jpuk osQ pj.k%
1. BC ls 'kh"kZ A  dh nwljh vksj U;wudks.k cukrh gqbZ ,d fdj.k BX  [khafp,A
2. BX ij 4 fcanq (
3
4
 esa 3 vkSj 4 esa ls cM+h la[;k) B
1
, B
2
, B
3
 vkSj B
4
, bl izdkj
vafdr dhft, fd BB
1
 = B
1
B
2
 = B
2
B
3
 = B
3
B
4 
gksA
3. B
4
C feykb, vkSj B
3
 (rhljs ¯cnq] ;gk¡
3
4
 esa 3 vkSj 4 esa ls 3 NksVh gS) ls
gksdj tkus okyh B
4
C osQ lekarj ,d js[kk
BC dks C'  ij izfrPNsn djrh gqbZ [khafp,A
4. C' ls gksdj tkus okyh CA  osQ lekarj
,d js[kk BA dks A' ij izfrPNsn djrh
gqbZ [khafp, (nsf[k, vko`Qfr 11-3)A
rc] ? A'BC' vHkh"V f=kHkqt gSA
jpuk,¡ 241
vkb, ns[ksa fd bl jpuk ls oSQls vHkh"V f=kHkqt izkIr gks tkrk gSA
jpuk 11-1 ls]  
BC 3
CC 1
'
=
'
blfy,] 
BC BC + C C C C 1 4
1 1
BC BC BC 3 3
' ' '
= = + = + =
' ' '
, vFkkZr~ 
BC
BC
'
 = 
3
4
 gSA
lkFk gh] C'A', CA osQ lekarj gSA blfy,  ? A'BC' ~ ? ABC (D;ksa\)
vr%]
A B A C BC 3
AB AC BC 4
' ' ' '
= = =
mnkgj.k 2 : ,d fn, x, f=kHkqt  ABC osQ le:i ,d f=kHkqt dh jpuk dhft,]
ftldh Hkqtk,¡ f=kHkqt ABC dh laxr Hkqtkvksa dh 
5
3
 gksa (vFkkZr~ LosQy xq.kd 
5
3
 gS)A
gy : ,d f=kHkqt ABC fn;k x;k gSA gesa ,d f=kHkqt dh jpuk djuh gS] ftldh Hkqtk,¡
? ABC dh laxr Hkqtkvksa dh 
5
3
 gksaA
jpuk osQ pj.k%
1. BC ls 'kh"kZ A osQ nwljh vksj U;wudks.k cukrh gqbZ ,d fdj.k BX [khafp,A
2. 5 (
5
3
 esa 5 vkSj 3 esa ls cM+h la[;k) ¯cnq B
1
, B
2
, B
3
, B
4
 vkSj B
5
, BX ij bl izdkj
vafdr dhft, fd BB
1
 = B
1
B
2
 = B
2
B
3
 = B
3
B
4
 = B
4
B
5 
gksA
3. B
3
 (rhljk ¯cnq] 
5
3
 esa 5 vkSj 3 esa ls NksVh la[;k) dks C ls feykb, vkSj B
5
 ls
gksdj tkus okyh  B
3
C osQ lekarj ,d js[kk] c<+k, x, js[kk[kaM BC dks C' ij
izfrPNsn djrh gqbZ [khafp,A
4. C' ls gksdj tkus okyh CA osQ lekarj ,d
js[kk] c<+kus ij js[kk[kaM BA dks A' ij izfrPNsn
djrh gqbZ [khafp, (nsf[k, vko`Qfr 11-4)A
rc] A'BC' vHkh"V f=kHkqt gSA
jpuk osQ vkSfpR; fl¼ djus osQ fy,]
è;ku nhft,  ? ABC ~ ? A'BC' (D;ksa\)
blfy, 
AB AC BC
AB AC BC
= =
' ' ' '
 gSA
vko`Qfr 11.4
242 xf.kr
ijarq
3
5
BB BC 3
BC BB 5
= =
'
 gSA
blfy,
BC 5
BC 3
'
= gS vkSj blhfy,
A B A C BC 5
AB AC BC 3
' ' ' '
= = =
gSA
fVIi.kh : mnkgj.k 1 vkSj 2 esa vki AB vFkok AC ls U;wudks.k cukrh gqbZ fdj.k Hkh
ys ldrs Fks vkSj mlh izdkj vkxs c<+ ldrs FksA
iz'ukoyh 11.1
fuEu esa ls izR;sd osQ fy, jpuk dk vkSfpR; Hkh nhft,%
1. 7.6 cm yack ,d js[kk[kaM [khafp, vkSj bls 5 : 8 vuqikr esa foHkkftr dhft,A nksuksa Hkkxksa dks
ekfi,A
2. 4 cm, 5 cm vkSj  6 cm Hkqtkvksa okys ,d f=kHkqt dh jpuk dhft, vkSj fiQj blosQ le:i ,d
vU; f=kHkqt dh jpuk dhft,] ftldh Hkqtk,¡ fn, gq, f=kHkqt dh laxr Hkqtkvksa dh 
2
3
 xquh
gksaA
3. 5 cm, 6 cm vkSj  7 cm Hkqtkvksa okys ,d f=kHkqt dh jpuk dhft, vkSj fiQj ,d vU; f=kHkqt
dh jpuk dhft,] ftldh Hkqtk,¡ fn;s gq,s f=kHkqt dh laxr Hkqtkvksa dh 
7
5
 xquh gksaA
4. vk/kj  8 cm rFkk Å¡pkbZ  4 cm osQ ,d lef}ckgq f=kHkqt dh jpuk dhft, vkSj fiQj ,d vU;
f=kHkqt dh jpuk dhft,] ftldh Hkqtk,¡ bl lef}ckgq f=kHkqt dh laxr Hkqtkvksa dh 
1
1
2
 xquh
gksaA
5. ,d f=kHkqt  ABC cukb, ftlesa  BC = 6 cm, AB = 5 cm  vkSj  ? ABC = 60° gksA fiQj ,d f=kHkqt
dh jpuk dhft,] ftldh Hkqtk,¡ ? ABC  dh laxr Hkqtkvksa dh  
3
4
 xquh gksaA
6. ,d f=kHkqt ABC cukb,]  ftlesa  BC = 7 cm, ? B = 45°, ? A = 105°  gksA fiQj ,d vU; f=kHkqt
dh jpuk dhft,] ftldh Hkqtk,¡ ? ABC  dh laxr Hkqtkvksa dh 
4
3
 xquh gksaA
7. ,d ledks.k f=kHkqt dh jpuk dhft,] ftldh Hkqtk,¡ (d.kZ osQ vfrfjDr) 4 cm rFkk  3 cm
yackbZ dh gksaA fiQj ,d vU; f=kHkqt dh jpuk dhft,] ftldh Hkqtk,¡ fn, gq, f=kHkqt dh
laxr Hkqtkvksa dh  
5
3
 xquh gksaA
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