NCERT पाठ्यपुस्तक पाठ 6 - त्रिभुज, कक्षा 10, गणित Class 10 Notes | EduRev

गणित कक्षा 10

Class 10 : NCERT पाठ्यपुस्तक पाठ 6 - त्रिभुज, कक्षा 10, गणित Class 10 Notes | EduRev

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130 xf.kr
6
6.1 Hkwfedk
vki viuh fiNyh d{kkvksa ls] f=kHkqtksa vkSj muosQ vusd xq.k/eks± ls Hkyh Hkk¡fr ifjfpr
gSaA d{kk IX esa] vki f=kHkqtksa dh lok±xlerk osQ ckjs esa foLr`r :i ls vè;;u dj pqosQ
gSaA ;kn dhft, fd nks f=kHkqt lok±xle rc dgs tkrs gSa tc muosQ leku vkdkj (shape)
rFkk leku vkeki (size) gksaA bl vè;k; esa] ge ,slh vko`Qfr;ksa osQ ckjs esa vè;;u djsaxs
ftuosQ vkdkj leku gksa ijarq muosQ vkeki dk leku gksuk vko';d ugha gksA nks vko`Qfr;k¡
ftuosQ leku vkdkj gksa (ijarq leku vkeki gksuk vko';d u gks) le:i vko`Qfr;k¡
(similar figures) dgykrh gSaA fo'ks"k :i ls] ge le:i f=kHkqtksa dh ppkZ djsaxs rFkk bl
tkudkjh dks igys i<+h xbZ ikbFkkxksjl izes; dh ,d ljy miifÙk nsus esa iz;ksx djsaxsA
f=kHkqt
2018-19
Page 2


130 xf.kr
6
6.1 Hkwfedk
vki viuh fiNyh d{kkvksa ls] f=kHkqtksa vkSj muosQ vusd xq.k/eks± ls Hkyh Hkk¡fr ifjfpr
gSaA d{kk IX esa] vki f=kHkqtksa dh lok±xlerk osQ ckjs esa foLr`r :i ls vè;;u dj pqosQ
gSaA ;kn dhft, fd nks f=kHkqt lok±xle rc dgs tkrs gSa tc muosQ leku vkdkj (shape)
rFkk leku vkeki (size) gksaA bl vè;k; esa] ge ,slh vko`Qfr;ksa osQ ckjs esa vè;;u djsaxs
ftuosQ vkdkj leku gksa ijarq muosQ vkeki dk leku gksuk vko';d ugha gksA nks vko`Qfr;k¡
ftuosQ leku vkdkj gksa (ijarq leku vkeki gksuk vko';d u gks) le:i vko`Qfr;k¡
(similar figures) dgykrh gSaA fo'ks"k :i ls] ge le:i f=kHkqtksa dh ppkZ djsaxs rFkk bl
tkudkjh dks igys i<+h xbZ ikbFkkxksjl izes; dh ,d ljy miifÙk nsus esa iz;ksx djsaxsA
f=kHkqt
2018-19
f=kHkqt 131
D;k vki vuqeku yxk ldrs gSa fd ioZrksa (tSls ekÅaV ,ojsLV) dh Å¡pkbZ;k¡ vFkok
oqQN nwjLFk oLrqvksa (tSls pUnzek) dh nwfj;k¡ fdl izdkj Kkr dh xbZ gSa\ D;k vki lksprs
gSa fd bUgsa ,d ekius okys iQhrs ls lh/k (izR;{k) ekik x;k gS\ okLro esa] bu lHkh
Å¡pkbZ vkSj nwfj;ksa dks vizR;{k ekiu (indirect measurement) dh vo/kj.kk dk iz;ksx
djrs gq, Kkr fd;k x;k gS] tks vko`Qfr;ksa dh le:irk osQ fl¼kar ij vk/kfjr gS (nsf[k,
mnkgj.k 7] iz'ukoyh 6-3 dk iz'u 15 rFkk lkFk gh bl iqLrd osQ vè;k; 8 vkSj 9)A
6.2 le:i vko`Qfr;k¡
d{kk IX  esa] vkius ns[kk Fkk fd leku
(,d gh) f=kT;k okys lHkh o`Ùk lok±xle
gksrs gSa] leku yackbZ dh Hkqtk okys lHkh
oxZ lok±xle gksrs gSa rFkk leku yackbZ
dh Hkqtk okys lHkh leckgq f=kHkqt
lok±xle gksrs gSaA
vc fdUgha nks (;k vf/d)
o`Ùkksa ij fopkj dhft, [nsf[k,
vko`Qfr 6.1 (i)]A D;k ;s lok±xle gSa\
pw¡fd buesa ls lHkh dh f=kT;k leku
ugha gS] blfy, ;s ijLij lok±xle
ugha gSaA è;ku nhft, fd buesa oqQN
lok±xle gSa vkSj oqQN lok±xle ugha gSa]
ijarq buesa ls lHkh osQ vkdkj leku gSaA
vr%] ;s lHkh os vko`Qfr;k¡ gSa ftUgsa ge
le:i (similar) dgrs gSaA nks le:i
vko`Qfr;ksa osQ vkdkj leku gksrs gSa ijarq
buosQ vkeki leku gksus vko';d ugha
gSaA vr%] lHkh o`Ùk le:i gksrs gSaA nks
(;k vf/d) oxks± osQ ckjs esa vFkok nks
(;k vf/d) leckgq f=kHkqtksa osQ ckjs esa vki D;k lksprs gSa [nsf[k, vko`Qfr 6.1 (ii)
vkSj (iii)]? lHkh o`Ùkkas dh rjg gh] ;gk¡ lHkh oxZ le:i gSa rFkk lHkh leckgq f=kHkqt
le:i gSaA
mijksDr ppkZ ls] ge ;g Hkh dg ldrs gSa fd lHkh lok±xle vko`Qfr;k¡ le:i
gksrh gSa] ijarq lHkh le:i vko`Qfr;ksa dk lok±xle gksuk vko';d ugha gSA
vko`Qfr 6.1
2018-19
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130 xf.kr
6
6.1 Hkwfedk
vki viuh fiNyh d{kkvksa ls] f=kHkqtksa vkSj muosQ vusd xq.k/eks± ls Hkyh Hkk¡fr ifjfpr
gSaA d{kk IX esa] vki f=kHkqtksa dh lok±xlerk osQ ckjs esa foLr`r :i ls vè;;u dj pqosQ
gSaA ;kn dhft, fd nks f=kHkqt lok±xle rc dgs tkrs gSa tc muosQ leku vkdkj (shape)
rFkk leku vkeki (size) gksaA bl vè;k; esa] ge ,slh vko`Qfr;ksa osQ ckjs esa vè;;u djsaxs
ftuosQ vkdkj leku gksa ijarq muosQ vkeki dk leku gksuk vko';d ugha gksA nks vko`Qfr;k¡
ftuosQ leku vkdkj gksa (ijarq leku vkeki gksuk vko';d u gks) le:i vko`Qfr;k¡
(similar figures) dgykrh gSaA fo'ks"k :i ls] ge le:i f=kHkqtksa dh ppkZ djsaxs rFkk bl
tkudkjh dks igys i<+h xbZ ikbFkkxksjl izes; dh ,d ljy miifÙk nsus esa iz;ksx djsaxsA
f=kHkqt
2018-19
f=kHkqt 131
D;k vki vuqeku yxk ldrs gSa fd ioZrksa (tSls ekÅaV ,ojsLV) dh Å¡pkbZ;k¡ vFkok
oqQN nwjLFk oLrqvksa (tSls pUnzek) dh nwfj;k¡ fdl izdkj Kkr dh xbZ gSa\ D;k vki lksprs
gSa fd bUgsa ,d ekius okys iQhrs ls lh/k (izR;{k) ekik x;k gS\ okLro esa] bu lHkh
Å¡pkbZ vkSj nwfj;ksa dks vizR;{k ekiu (indirect measurement) dh vo/kj.kk dk iz;ksx
djrs gq, Kkr fd;k x;k gS] tks vko`Qfr;ksa dh le:irk osQ fl¼kar ij vk/kfjr gS (nsf[k,
mnkgj.k 7] iz'ukoyh 6-3 dk iz'u 15 rFkk lkFk gh bl iqLrd osQ vè;k; 8 vkSj 9)A
6.2 le:i vko`Qfr;k¡
d{kk IX  esa] vkius ns[kk Fkk fd leku
(,d gh) f=kT;k okys lHkh o`Ùk lok±xle
gksrs gSa] leku yackbZ dh Hkqtk okys lHkh
oxZ lok±xle gksrs gSa rFkk leku yackbZ
dh Hkqtk okys lHkh leckgq f=kHkqt
lok±xle gksrs gSaA
vc fdUgha nks (;k vf/d)
o`Ùkksa ij fopkj dhft, [nsf[k,
vko`Qfr 6.1 (i)]A D;k ;s lok±xle gSa\
pw¡fd buesa ls lHkh dh f=kT;k leku
ugha gS] blfy, ;s ijLij lok±xle
ugha gSaA è;ku nhft, fd buesa oqQN
lok±xle gSa vkSj oqQN lok±xle ugha gSa]
ijarq buesa ls lHkh osQ vkdkj leku gSaA
vr%] ;s lHkh os vko`Qfr;k¡ gSa ftUgsa ge
le:i (similar) dgrs gSaA nks le:i
vko`Qfr;ksa osQ vkdkj leku gksrs gSa ijarq
buosQ vkeki leku gksus vko';d ugha
gSaA vr%] lHkh o`Ùk le:i gksrs gSaA nks
(;k vf/d) oxks± osQ ckjs esa vFkok nks
(;k vf/d) leckgq f=kHkqtksa osQ ckjs esa vki D;k lksprs gSa [nsf[k, vko`Qfr 6.1 (ii)
vkSj (iii)]? lHkh o`Ùkkas dh rjg gh] ;gk¡ lHkh oxZ le:i gSa rFkk lHkh leckgq f=kHkqt
le:i gSaA
mijksDr ppkZ ls] ge ;g Hkh dg ldrs gSa fd lHkh lok±xle vko`Qfr;k¡ le:i
gksrh gSa] ijarq lHkh le:i vko`Qfr;ksa dk lok±xle gksuk vko';d ugha gSA
vko`Qfr 6.1
2018-19
132 xf.kr
D;k ,d o`Ùk vkSj ,d oxZ le:i gks ldrs gSa\ D;k ,d f=kHkqt vkSj ,d oxZ
le:i gks ldrs gSa\ bu vko`Qfr;ksa dks ns[kus ek=k ls gh vki iz'uksa osQ mÙkj ns ldrs gSa
(nsf[k, vko`Qfr 6-1)A Li"V 'kCnksa esa] ;s vko`Qfr;k¡ le:i ugha gSaA (D;ksa\)
vki nks prqHkqZtksa ABCD vkSj PQRS
osQ ckjs esa D;k dg ldrs gSa (nsf[k,
vko`Qfr 6-2)\ D;k ;s le:i gSa\ ;s
vko`Qfr;k¡ le:i&lh izrhr gks jgh gSa]
ijarq ge blosQ ckjs esa fuf'pr :i ls
oqQN ugha dg ldrsA blfy,] ;g
vko';d gks tkrk gS fd ge vko`Qfr;ksa dh le:irk osQ fy, dksbZ ifjHkk"kk Kkr djsa rFkk
bl ifjHkk"kk ij vk/kfjr ;g lqfuf'pr djus osQ fy, fd nks nh gqbZ vko`Qfr;k¡ le:i
gSa ;k ugha] oqQN fu;e izkIr djsaA blosQ fy,] vkb, vko`Qfr 6-3 esa fp=kksa dks ns[ksa%
vko`Qfr 6.3
vki rqjar ;g dgsaxs fd ;s ,d gh Lekjd (rktegy) osQ fp=k gSa] ijarq ;s
fHkUu&fHkUu vkekiksa (sizes) osQ gSaA D;k vki ;g dgsaxs fd ;s fp=k le:i gSa\ gk¡] ;s gSaA
vki ,d gh O;fDr osQ ,d gh vkeki okys mu nks fp=kksa osQ ckjs esa D;k dg ldrs gSa]
ftuesa ls ,d mldh 10 o"kZ dh vk;q dk gS rFkk nwljk mldh 40 o"kZ dh vk;q dk gS\
D;k ;s nksuksa fp=k le:i gSa\ ;s fp=k leku vkeki osQ gSa] ijarq fuf'pr :i ls ;s leku
vkdkj osQ ugha gSaA vr%] ;s le:i ugha gSaA
tc dksbZ i
+
QksVksxzkiQj ,d gh usxsfVo ls fofHkUu ekiksa osQ i
+
QksVks ¯izV fudkyrh gS] rks og
D;k djrh gS\ vkius LVSai lkbT
+
k] ikliksVZ lkbT
+
k ,oa iksLV dkMZ lkbT
+
k i
+
QksVks (;k fp=kksa)
osQ ckjs esa vo'; lquk gksxkA og lkekU; :i ls ,d NksVs vkeki (lkbt) dh fi
+
QYe
(film), eku yhft, tks 35 mm vkeki okyh fi
+
QYe gS] ij i
+
QksVks [khaprh gS vkSj fiQj mls
,d cM+s vkeki] tSls 45 mm (;k 55 mm) vkeki] okyh i
+
QksVks osQ :i esa vko£/r
vko`Qfr 6.2
2018-19
Page 4


130 xf.kr
6
6.1 Hkwfedk
vki viuh fiNyh d{kkvksa ls] f=kHkqtksa vkSj muosQ vusd xq.k/eks± ls Hkyh Hkk¡fr ifjfpr
gSaA d{kk IX esa] vki f=kHkqtksa dh lok±xlerk osQ ckjs esa foLr`r :i ls vè;;u dj pqosQ
gSaA ;kn dhft, fd nks f=kHkqt lok±xle rc dgs tkrs gSa tc muosQ leku vkdkj (shape)
rFkk leku vkeki (size) gksaA bl vè;k; esa] ge ,slh vko`Qfr;ksa osQ ckjs esa vè;;u djsaxs
ftuosQ vkdkj leku gksa ijarq muosQ vkeki dk leku gksuk vko';d ugha gksA nks vko`Qfr;k¡
ftuosQ leku vkdkj gksa (ijarq leku vkeki gksuk vko';d u gks) le:i vko`Qfr;k¡
(similar figures) dgykrh gSaA fo'ks"k :i ls] ge le:i f=kHkqtksa dh ppkZ djsaxs rFkk bl
tkudkjh dks igys i<+h xbZ ikbFkkxksjl izes; dh ,d ljy miifÙk nsus esa iz;ksx djsaxsA
f=kHkqt
2018-19
f=kHkqt 131
D;k vki vuqeku yxk ldrs gSa fd ioZrksa (tSls ekÅaV ,ojsLV) dh Å¡pkbZ;k¡ vFkok
oqQN nwjLFk oLrqvksa (tSls pUnzek) dh nwfj;k¡ fdl izdkj Kkr dh xbZ gSa\ D;k vki lksprs
gSa fd bUgsa ,d ekius okys iQhrs ls lh/k (izR;{k) ekik x;k gS\ okLro esa] bu lHkh
Å¡pkbZ vkSj nwfj;ksa dks vizR;{k ekiu (indirect measurement) dh vo/kj.kk dk iz;ksx
djrs gq, Kkr fd;k x;k gS] tks vko`Qfr;ksa dh le:irk osQ fl¼kar ij vk/kfjr gS (nsf[k,
mnkgj.k 7] iz'ukoyh 6-3 dk iz'u 15 rFkk lkFk gh bl iqLrd osQ vè;k; 8 vkSj 9)A
6.2 le:i vko`Qfr;k¡
d{kk IX  esa] vkius ns[kk Fkk fd leku
(,d gh) f=kT;k okys lHkh o`Ùk lok±xle
gksrs gSa] leku yackbZ dh Hkqtk okys lHkh
oxZ lok±xle gksrs gSa rFkk leku yackbZ
dh Hkqtk okys lHkh leckgq f=kHkqt
lok±xle gksrs gSaA
vc fdUgha nks (;k vf/d)
o`Ùkksa ij fopkj dhft, [nsf[k,
vko`Qfr 6.1 (i)]A D;k ;s lok±xle gSa\
pw¡fd buesa ls lHkh dh f=kT;k leku
ugha gS] blfy, ;s ijLij lok±xle
ugha gSaA è;ku nhft, fd buesa oqQN
lok±xle gSa vkSj oqQN lok±xle ugha gSa]
ijarq buesa ls lHkh osQ vkdkj leku gSaA
vr%] ;s lHkh os vko`Qfr;k¡ gSa ftUgsa ge
le:i (similar) dgrs gSaA nks le:i
vko`Qfr;ksa osQ vkdkj leku gksrs gSa ijarq
buosQ vkeki leku gksus vko';d ugha
gSaA vr%] lHkh o`Ùk le:i gksrs gSaA nks
(;k vf/d) oxks± osQ ckjs esa vFkok nks
(;k vf/d) leckgq f=kHkqtksa osQ ckjs esa vki D;k lksprs gSa [nsf[k, vko`Qfr 6.1 (ii)
vkSj (iii)]? lHkh o`Ùkkas dh rjg gh] ;gk¡ lHkh oxZ le:i gSa rFkk lHkh leckgq f=kHkqt
le:i gSaA
mijksDr ppkZ ls] ge ;g Hkh dg ldrs gSa fd lHkh lok±xle vko`Qfr;k¡ le:i
gksrh gSa] ijarq lHkh le:i vko`Qfr;ksa dk lok±xle gksuk vko';d ugha gSA
vko`Qfr 6.1
2018-19
132 xf.kr
D;k ,d o`Ùk vkSj ,d oxZ le:i gks ldrs gSa\ D;k ,d f=kHkqt vkSj ,d oxZ
le:i gks ldrs gSa\ bu vko`Qfr;ksa dks ns[kus ek=k ls gh vki iz'uksa osQ mÙkj ns ldrs gSa
(nsf[k, vko`Qfr 6-1)A Li"V 'kCnksa esa] ;s vko`Qfr;k¡ le:i ugha gSaA (D;ksa\)
vki nks prqHkqZtksa ABCD vkSj PQRS
osQ ckjs esa D;k dg ldrs gSa (nsf[k,
vko`Qfr 6-2)\ D;k ;s le:i gSa\ ;s
vko`Qfr;k¡ le:i&lh izrhr gks jgh gSa]
ijarq ge blosQ ckjs esa fuf'pr :i ls
oqQN ugha dg ldrsA blfy,] ;g
vko';d gks tkrk gS fd ge vko`Qfr;ksa dh le:irk osQ fy, dksbZ ifjHkk"kk Kkr djsa rFkk
bl ifjHkk"kk ij vk/kfjr ;g lqfuf'pr djus osQ fy, fd nks nh gqbZ vko`Qfr;k¡ le:i
gSa ;k ugha] oqQN fu;e izkIr djsaA blosQ fy,] vkb, vko`Qfr 6-3 esa fp=kksa dks ns[ksa%
vko`Qfr 6.3
vki rqjar ;g dgsaxs fd ;s ,d gh Lekjd (rktegy) osQ fp=k gSa] ijarq ;s
fHkUu&fHkUu vkekiksa (sizes) osQ gSaA D;k vki ;g dgsaxs fd ;s fp=k le:i gSa\ gk¡] ;s gSaA
vki ,d gh O;fDr osQ ,d gh vkeki okys mu nks fp=kksa osQ ckjs esa D;k dg ldrs gSa]
ftuesa ls ,d mldh 10 o"kZ dh vk;q dk gS rFkk nwljk mldh 40 o"kZ dh vk;q dk gS\
D;k ;s nksuksa fp=k le:i gSa\ ;s fp=k leku vkeki osQ gSa] ijarq fuf'pr :i ls ;s leku
vkdkj osQ ugha gSaA vr%] ;s le:i ugha gSaA
tc dksbZ i
+
QksVksxzkiQj ,d gh usxsfVo ls fofHkUu ekiksa osQ i
+
QksVks ¯izV fudkyrh gS] rks og
D;k djrh gS\ vkius LVSai lkbT
+
k] ikliksVZ lkbT
+
k ,oa iksLV dkMZ lkbT
+
k i
+
QksVks (;k fp=kksa)
osQ ckjs esa vo'; lquk gksxkA og lkekU; :i ls ,d NksVs vkeki (lkbt) dh fi
+
QYe
(film), eku yhft, tks 35 mm vkeki okyh fi
+
QYe gS] ij i
+
QksVks [khaprh gS vkSj fiQj mls
,d cM+s vkeki] tSls 45 mm (;k 55 mm) vkeki] okyh i
+
QksVks osQ :i esa vko£/r
vko`Qfr 6.2
2018-19
f=kHkqt 133
djrh gSA bl izdkj] ;fn ge NksVs fp=k osQ fdlh ,d js[kk[kaM dks ysa] rks cM+s fp=k esa
bldk laxr js[kk[kaM] yackbZ esa igys js[kk[kaM dk 
4 5
35
;k
55
35
?
?
?
?
?
? xquk gksxkA okLro esa
bldk vFkZ ;g gS fd NksVs fp=k dk izR;sd js[kk[kaM 35%45 (;k 35%55) osQ vuqikr esa
vko£/r gks (c<+) x;k gSA blh dks bl izdkj Hkh dgk tk ldrk gS fd cM+s fp=k dk
izR;sd js[kk[kaM 45%35 (;k 55%35) osQ vuqikr esa ?kV (de gks) x;k gSA lkFk gh] ;fn
vki fofHkUu vkekiksa osQ nks fp=kksa esa laxr js[kk[kaMksa osQ fdlh Hkh ;qXe osQ chp cus >qdkoksa
¹vFkok dks.kksaº dks ysa] rks vki ns[ksaxs fd ;s >qdko (;k dks.k) lnSo cjkcj gksaxsA ;gh
nks vko`Qfr;ksa rFkk fo'ks"kdj nks cgqHkqtksa dh le:irk dk lkj gSA ge dgrs gSa fd%
Hkqtkvksa dh leku la[;k okys nks cgqHkqt le:i gksrs gSa] ;fn  (i) muosQ laxr dks.k
cjkcj gksa rFkk (ii) budh laxr Hkqtk,¡ ,d gh vuqikr esa (vFkkZr~ lekuqikrh) gksaA
è;ku nhft, fd cgqHkqtksa osQ fy, laxr Hkqtkvksa osQ bl ,d gh vuqikr dks LosQy
xq.kd (scale factor) ¹vFkok izfrfuf/Ro fHkUu (Representative Fraction)º dgk tkrk
gSA vkius ;g vo'; lquk gksxk fd fo'o ekufp=k ¹vFkkZr~ Xykscy ekufp=kº rFkk Hkouksa
osQ fuekZ.k osQ fy, cuk, tkus okyh :i js[kk ,d mi;qDr LosQy xq.kd rFkk oqQN ifjikfV;ksa
dks è;ku esa j[kdj cuk, tkrs gSaA
vko`Qfr;ksa dh le:irk dks vf/d Li"V :i ls le>us osQ fy,] vkb, fuEufyf[kr
fØ;kdyki djsa%
fØ;kdyki 1 : viuh d{kk osQ dejs dh
Nr osQ fdlh ¯cnq O ij izdk'k ;qDr cYc
yxkb, rFkk mlosQ Bhd uhps ,d est jf[k,A
vkb, ,d lery dkMZcksMZ esa ls ,d cgqHkqt]
eku yhft, prqHkqZt ABCD, dkV ysa rFkk bl
dkMZcksMZ dks Hkwfe osQ lekarj est vkSj tyrs
gq, cYc osQ chp esa j[ksaA rc] est ij
ABCD dh ,d Nk;k (shadow) iM+sxhA bl
Nk;k dh ckgjh :ijs[kk dks A'B'C' D' ls fpfãr
dhft, (nsf[k, vko`Qfr 6-4)A
è;ku nhft, fd prqHkqZt A'B'C'D' prqHkqZt
vko`Qfr 6.4
2018-19
Page 5


130 xf.kr
6
6.1 Hkwfedk
vki viuh fiNyh d{kkvksa ls] f=kHkqtksa vkSj muosQ vusd xq.k/eks± ls Hkyh Hkk¡fr ifjfpr
gSaA d{kk IX esa] vki f=kHkqtksa dh lok±xlerk osQ ckjs esa foLr`r :i ls vè;;u dj pqosQ
gSaA ;kn dhft, fd nks f=kHkqt lok±xle rc dgs tkrs gSa tc muosQ leku vkdkj (shape)
rFkk leku vkeki (size) gksaA bl vè;k; esa] ge ,slh vko`Qfr;ksa osQ ckjs esa vè;;u djsaxs
ftuosQ vkdkj leku gksa ijarq muosQ vkeki dk leku gksuk vko';d ugha gksA nks vko`Qfr;k¡
ftuosQ leku vkdkj gksa (ijarq leku vkeki gksuk vko';d u gks) le:i vko`Qfr;k¡
(similar figures) dgykrh gSaA fo'ks"k :i ls] ge le:i f=kHkqtksa dh ppkZ djsaxs rFkk bl
tkudkjh dks igys i<+h xbZ ikbFkkxksjl izes; dh ,d ljy miifÙk nsus esa iz;ksx djsaxsA
f=kHkqt
2018-19
f=kHkqt 131
D;k vki vuqeku yxk ldrs gSa fd ioZrksa (tSls ekÅaV ,ojsLV) dh Å¡pkbZ;k¡ vFkok
oqQN nwjLFk oLrqvksa (tSls pUnzek) dh nwfj;k¡ fdl izdkj Kkr dh xbZ gSa\ D;k vki lksprs
gSa fd bUgsa ,d ekius okys iQhrs ls lh/k (izR;{k) ekik x;k gS\ okLro esa] bu lHkh
Å¡pkbZ vkSj nwfj;ksa dks vizR;{k ekiu (indirect measurement) dh vo/kj.kk dk iz;ksx
djrs gq, Kkr fd;k x;k gS] tks vko`Qfr;ksa dh le:irk osQ fl¼kar ij vk/kfjr gS (nsf[k,
mnkgj.k 7] iz'ukoyh 6-3 dk iz'u 15 rFkk lkFk gh bl iqLrd osQ vè;k; 8 vkSj 9)A
6.2 le:i vko`Qfr;k¡
d{kk IX  esa] vkius ns[kk Fkk fd leku
(,d gh) f=kT;k okys lHkh o`Ùk lok±xle
gksrs gSa] leku yackbZ dh Hkqtk okys lHkh
oxZ lok±xle gksrs gSa rFkk leku yackbZ
dh Hkqtk okys lHkh leckgq f=kHkqt
lok±xle gksrs gSaA
vc fdUgha nks (;k vf/d)
o`Ùkksa ij fopkj dhft, [nsf[k,
vko`Qfr 6.1 (i)]A D;k ;s lok±xle gSa\
pw¡fd buesa ls lHkh dh f=kT;k leku
ugha gS] blfy, ;s ijLij lok±xle
ugha gSaA è;ku nhft, fd buesa oqQN
lok±xle gSa vkSj oqQN lok±xle ugha gSa]
ijarq buesa ls lHkh osQ vkdkj leku gSaA
vr%] ;s lHkh os vko`Qfr;k¡ gSa ftUgsa ge
le:i (similar) dgrs gSaA nks le:i
vko`Qfr;ksa osQ vkdkj leku gksrs gSa ijarq
buosQ vkeki leku gksus vko';d ugha
gSaA vr%] lHkh o`Ùk le:i gksrs gSaA nks
(;k vf/d) oxks± osQ ckjs esa vFkok nks
(;k vf/d) leckgq f=kHkqtksa osQ ckjs esa vki D;k lksprs gSa [nsf[k, vko`Qfr 6.1 (ii)
vkSj (iii)]? lHkh o`Ùkkas dh rjg gh] ;gk¡ lHkh oxZ le:i gSa rFkk lHkh leckgq f=kHkqt
le:i gSaA
mijksDr ppkZ ls] ge ;g Hkh dg ldrs gSa fd lHkh lok±xle vko`Qfr;k¡ le:i
gksrh gSa] ijarq lHkh le:i vko`Qfr;ksa dk lok±xle gksuk vko';d ugha gSA
vko`Qfr 6.1
2018-19
132 xf.kr
D;k ,d o`Ùk vkSj ,d oxZ le:i gks ldrs gSa\ D;k ,d f=kHkqt vkSj ,d oxZ
le:i gks ldrs gSa\ bu vko`Qfr;ksa dks ns[kus ek=k ls gh vki iz'uksa osQ mÙkj ns ldrs gSa
(nsf[k, vko`Qfr 6-1)A Li"V 'kCnksa esa] ;s vko`Qfr;k¡ le:i ugha gSaA (D;ksa\)
vki nks prqHkqZtksa ABCD vkSj PQRS
osQ ckjs esa D;k dg ldrs gSa (nsf[k,
vko`Qfr 6-2)\ D;k ;s le:i gSa\ ;s
vko`Qfr;k¡ le:i&lh izrhr gks jgh gSa]
ijarq ge blosQ ckjs esa fuf'pr :i ls
oqQN ugha dg ldrsA blfy,] ;g
vko';d gks tkrk gS fd ge vko`Qfr;ksa dh le:irk osQ fy, dksbZ ifjHkk"kk Kkr djsa rFkk
bl ifjHkk"kk ij vk/kfjr ;g lqfuf'pr djus osQ fy, fd nks nh gqbZ vko`Qfr;k¡ le:i
gSa ;k ugha] oqQN fu;e izkIr djsaA blosQ fy,] vkb, vko`Qfr 6-3 esa fp=kksa dks ns[ksa%
vko`Qfr 6.3
vki rqjar ;g dgsaxs fd ;s ,d gh Lekjd (rktegy) osQ fp=k gSa] ijarq ;s
fHkUu&fHkUu vkekiksa (sizes) osQ gSaA D;k vki ;g dgsaxs fd ;s fp=k le:i gSa\ gk¡] ;s gSaA
vki ,d gh O;fDr osQ ,d gh vkeki okys mu nks fp=kksa osQ ckjs esa D;k dg ldrs gSa]
ftuesa ls ,d mldh 10 o"kZ dh vk;q dk gS rFkk nwljk mldh 40 o"kZ dh vk;q dk gS\
D;k ;s nksuksa fp=k le:i gSa\ ;s fp=k leku vkeki osQ gSa] ijarq fuf'pr :i ls ;s leku
vkdkj osQ ugha gSaA vr%] ;s le:i ugha gSaA
tc dksbZ i
+
QksVksxzkiQj ,d gh usxsfVo ls fofHkUu ekiksa osQ i
+
QksVks ¯izV fudkyrh gS] rks og
D;k djrh gS\ vkius LVSai lkbT
+
k] ikliksVZ lkbT
+
k ,oa iksLV dkMZ lkbT
+
k i
+
QksVks (;k fp=kksa)
osQ ckjs esa vo'; lquk gksxkA og lkekU; :i ls ,d NksVs vkeki (lkbt) dh fi
+
QYe
(film), eku yhft, tks 35 mm vkeki okyh fi
+
QYe gS] ij i
+
QksVks [khaprh gS vkSj fiQj mls
,d cM+s vkeki] tSls 45 mm (;k 55 mm) vkeki] okyh i
+
QksVks osQ :i esa vko£/r
vko`Qfr 6.2
2018-19
f=kHkqt 133
djrh gSA bl izdkj] ;fn ge NksVs fp=k osQ fdlh ,d js[kk[kaM dks ysa] rks cM+s fp=k esa
bldk laxr js[kk[kaM] yackbZ esa igys js[kk[kaM dk 
4 5
35
;k
55
35
?
?
?
?
?
? xquk gksxkA okLro esa
bldk vFkZ ;g gS fd NksVs fp=k dk izR;sd js[kk[kaM 35%45 (;k 35%55) osQ vuqikr esa
vko£/r gks (c<+) x;k gSA blh dks bl izdkj Hkh dgk tk ldrk gS fd cM+s fp=k dk
izR;sd js[kk[kaM 45%35 (;k 55%35) osQ vuqikr esa ?kV (de gks) x;k gSA lkFk gh] ;fn
vki fofHkUu vkekiksa osQ nks fp=kksa esa laxr js[kk[kaMksa osQ fdlh Hkh ;qXe osQ chp cus >qdkoksa
¹vFkok dks.kksaº dks ysa] rks vki ns[ksaxs fd ;s >qdko (;k dks.k) lnSo cjkcj gksaxsA ;gh
nks vko`Qfr;ksa rFkk fo'ks"kdj nks cgqHkqtksa dh le:irk dk lkj gSA ge dgrs gSa fd%
Hkqtkvksa dh leku la[;k okys nks cgqHkqt le:i gksrs gSa] ;fn  (i) muosQ laxr dks.k
cjkcj gksa rFkk (ii) budh laxr Hkqtk,¡ ,d gh vuqikr esa (vFkkZr~ lekuqikrh) gksaA
è;ku nhft, fd cgqHkqtksa osQ fy, laxr Hkqtkvksa osQ bl ,d gh vuqikr dks LosQy
xq.kd (scale factor) ¹vFkok izfrfuf/Ro fHkUu (Representative Fraction)º dgk tkrk
gSA vkius ;g vo'; lquk gksxk fd fo'o ekufp=k ¹vFkkZr~ Xykscy ekufp=kº rFkk Hkouksa
osQ fuekZ.k osQ fy, cuk, tkus okyh :i js[kk ,d mi;qDr LosQy xq.kd rFkk oqQN ifjikfV;ksa
dks è;ku esa j[kdj cuk, tkrs gSaA
vko`Qfr;ksa dh le:irk dks vf/d Li"V :i ls le>us osQ fy,] vkb, fuEufyf[kr
fØ;kdyki djsa%
fØ;kdyki 1 : viuh d{kk osQ dejs dh
Nr osQ fdlh ¯cnq O ij izdk'k ;qDr cYc
yxkb, rFkk mlosQ Bhd uhps ,d est jf[k,A
vkb, ,d lery dkMZcksMZ esa ls ,d cgqHkqt]
eku yhft, prqHkqZt ABCD, dkV ysa rFkk bl
dkMZcksMZ dks Hkwfe osQ lekarj est vkSj tyrs
gq, cYc osQ chp esa j[ksaA rc] est ij
ABCD dh ,d Nk;k (shadow) iM+sxhA bl
Nk;k dh ckgjh :ijs[kk dks A'B'C' D' ls fpfãr
dhft, (nsf[k, vko`Qfr 6-4)A
è;ku nhft, fd prqHkqZt A'B'C'D' prqHkqZt
vko`Qfr 6.4
2018-19
134 xf.kr
ABCD dk ,d vkdkj ifjo/Zu (;k vkoèkZu) gSA ;g izdk'k osQ bl xq.k/eZ osQ dkj.k
gS fd izdk'k lh/h js[kk esa pyrh gSA vki ;g Hkh ns[k ldrs gSa fd A' fdj.k OA ij fLFkr
gS] B' fdj.k OB ij fLFkr gS] C'  fdj.k  OC ij fLFkr gS rFkk D' fdj.k OD ij fLFkr gSA
bl izdkj] prqHkqZt A'B'C'D'  vkSj ABCD leku vkdkj osQ gSa_ ijarq buosQ eki fHkUu&fHkUu
gSaA
vr% prqHkqZt  A'B'C'D'  prqHkqZt ABCD osQ le:i gSA ge ;g Hkh dg ldrs gSa fd
prqHkqZt ABCD prqHkqZt A'B'C'D' osQ le:i gSA
;gk¡] vki ;g Hkh ns[k ldrs gSa fd 'kh"kZ A' 'kh"kZ A osQ laxr gS] 'kh"kZ B' 'kh"kZ B osQ
laxr gS] 'kh"kZ C' 'kh"kZ C osQ laxr gS rFkk 'kh"kZ D' 'kh"kZ D osQ laxr gSA lkaosQfrd :i ls bu
laxrrkvksa (correspondences) dks A ' ? A, B ' ? B , C' ? C vkSj D' ? D ls fu:fir
fd;k tkrk gSA nksuksa prqHkqZtksa osQ dks.kksa vkSj Hkqtkvksa dks okLrfod :i ls eki dj] vki
bldk lR;kiu dj ldrs gSa fd
(i) ? A = ? A', ? B = ? B', ? C = ? C', ? D = ? D' vkSj
(ii)  
AB B C CD DA
A B B C C D DA
= = =
' ' ' ' ' ' ''
.
blls iqu% ;g ckr Li"V gksrh gS fd Hkqtkvksa dh leku la[;k okys nks cgqHkqt
le:i gksrs gSa] ;fn (i) muosQ lHkh laxr dks.k cjkcj gksa rFkk (ii) mudh lHkh laxr Hkqtk,¡
,d gh vuqikr (lekuqikr) esa gksaA
mijskDr osQ vk/kj ij] vki ljyrk ls ;g dg ldrs gSa fd vko`Qfr 6-5 esa fn,
x, prqHkqZt ABCD vkSj PQRS le:i gSaA
vko`Qfr 6.5
2018-19
Read More
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