This document contains 27 multiple choice and long answer questions related to calculating areas of geometric shapes like triangles, quadrilaterals and circles. The questions test concepts like using properties of parallel lines, perpendicular lines, radii and angles to determine missing measures and calculate areas. Sample solutions are provided for 5 multiple choice questions as examples. The long answer section asks to prove that the areas of two shapes are equal.

1. 1
ul0ft z[ªvnf – !)
Hofldlt
kl/dflh{t ljlzi6Ls/0f tflnsf 2065 cg';f/ Hofldlt If]qaf6 4 cf]6f 5f]6f] pQ/ cfpg] 8 cªssf]
/ 4 cf]6f nfdf] pQ/ cfpg] 16 cªssf] u/L hDdf 24 cªssf 8 cf]6f k|Zgx¿ ;f]lwg] k|fjwfg
/x]sf] 5 . oxf“ k/LIffsf lgldQ pko'St / dxŒjk"0f{ dflgPsf ;d:ofx¿aLr pbfx/0f;lxt 5nkmn
ul/Psf] 5 .
-s_ 5f]6f] pQ/ cfpg] k|Zgx¿ M
lqe'hsf] If]qkmn M
pbfx/0f M 1
;“u}sf] lrqdf ABCD ;=r= xf] . AH=5cm, AHBC
/ BC=10cm eP ADE sf] If]qkmn lgsfNg'xf]; .
;dfwfg M oxf“, AH=5cm, BC=10cm
 ;=r= ABCD sf] If]qkmn
(1)
ca, ;=r=
[pxL cfwf/ / pxL ;dfgfGt/ /]vfx¿ ljrsf] ;=r= sf] If]qkmnsf] cfwf lqe'hsf] If]qkmn x'g]
x'gfn]] (1)
pbfx/0f M 2
;“u} lbOPsf] lrqdf O j[Qsf] s]Gb| laGb', olb ∡OPR = 350
eP ∡PSR sf] dfg slt xf]nf <
;dfwfg M ∡POR = 1800
– (2∡OPR)
= 1800
– 2 350
= 1800
– 700
= 1100
(1)
∡POR -a[xt_ = 3600
– 1100
= 2500
 ∡PSR = ∡POR -a[xt_ [Pp6} rfkdf cfwfl/t kl/lwsf]0f s]Gb|Lo sf]0fsf] cfwf x'gfn]]
=  2500
= 1250 (1)
OS
P
R
10 cm
A
B C
D
EH
5 cm

2. 2
pbfx/0f M 3
lbOPsf] lrqdf O s]Gb|laGb', eP
sf] dfg lgsfNg'xf]; .
;dfwfg M oxf“, / JK Jof; xf] .
-cw{j[Qdf aGg]sf]0f Ps ;dsf]0f x'G5 . _ (1)
-rj|mLo rt'e'{hsf] ;Dd'v sf]0fx¿sf] of]u b'O{ ;dsf]0f x'G5 ._ (1)
pbfx/0f M 4
;“u} lbOPsf] lrqdf, O j[Qsf] s]Gb|laGb', TAN :kz{
/]vf h;df A :kz{ laGb' 5 . olb ∡PRA = 640 eP
∡OSA slt x'G5 < lgsfNg'xf]; .
;dfwfg M lrqdf, AP Jof; / TAN :kz{ /]vf x'gfn],
∡PAR = 900
(
∡APR = 900
– ∡PRA
= 900
– 640
= 260
(1)
∡AOS = 2∡APR -Pp6} rfkdf cfwfl/t s]Gb|df ag]sf] sf]0f kl/lw sf]0fsf] bf]Aa/ x'gfn]_
= 2  260
= 520
 ∡OSA = 900
– 520
-;dsf]0f lqe'hsf] af“sL sf]0f_
= 380
(1)
pbfx/0f : 5
lrqdf AOB Jof; 5 . olb AB||CD / CEB=600
eP
BCD sf] gfk kQf nufpg'xf]; .
NT
O
P
A R S
J K
LI
? o
O
37o
E
A B
A
DC
O
600

3. 3
;dfwfg M lrqdf AOB Jof; / AB||CD lbOPsf] x'gfn], CAB=CEB=600
-pxL rfk BC df
ag]sf kl/lw sf]0fx¿_ ABC af6, ACB=900
-cw{j[Qdf ag]sf] kl/lwsf]0f ePsf]n]_ (1)
ABC = 1000
–900
–600
=300
BCD = ABC=300
-PsfGt/ sf]0fx¿ ePsf]n]_ (1)
pbfx/0f : 6
rj|mLo rt'e'{h ABCD df BCD = 740
/ AB=AD
eP ABD sf] gfk lgsfNg'xf]; .
;dfwfg M
BAD + BCD = 1800
-rj|mLo rt'e'{hsf] ;Dd'v sf]0fx¿sf] of]u b'O{ ;dsf]0f x'gfn]_
BAD = 1800
– 740
=1060
(1)
ABD af6,
BAD+ABD+ADB = 1800
-lqe'hsf sf]0fx¿sf] of]u_
or, ABD =ADB -;dlåe'h lqe'h ePsf]n]_
740
+ABD +ABD = 1800
or, 2ABD = 1060
ABD = 530
(1)
pbfx/0f : 7
lrqdf O s]Gb|, TAN :kz{ /]vf / A :kz{ laGb' 5 . olb
TAB = 400
eP ABO sf] gfk kQf nufpg'xf]; .
;dfwfg M OAT = 900
-:kz{ laGb' / s]Gb| laGb' hf]8bf :kz{ /]vfdf nDa x'G5 ._ (1)
 BAO = 900
–400
=500
OAB =OBA = 500
(OA = OB ePsfn]_ (1)
lbOPsf pbfx/0fx¿sf] ;xfotfn] tnsf ;d:ofx¿ ;dfwfg ug'{xf]; M
1. lbOPsf] lrqdf STR sf] If]qkmn eP ;=r=
PQRS sf] If]qkmn slt xf]nf <
pQ/ M 40 ju{ ;]=ld=
Q
P S T
R
A
B
D
C740
A
B
N
400
O
T

4. 4
2. lrqdf ABCD Pp6f ;de'h rt'e'{h / DCE ;/n /]vfv08 xf] .
olb AC=24cm / BD=30cm eP ABE sf] If]qkmn kQf
nufpg'xf]; . -pQ/ M 180cm2
_
3. PQRS Pp6f ju{ xf] / PST ;/n /]vfv08 xf] .
olb PR= eP TQR sf] If]qkmn kQf
nufpg'xf]; .
-pQ/ M 8cm2
_
4. ABCD Pp6f ;dnDa rt'e'{h xf] . olb
CDBC, AD=20cm, AB=16cm / BC=28cm
eP ABCD sf] If]qkmn kQf nufpg'xf]; .
-pQ/M 17.8cm2
_
5. lrqdf ABCD ;=r= / ADE Pp6f ;/n
/]vfv08 5 . BCE=36cm2
eP ABD sf]
If]qkmn kQf nufpg'xf]; . -pQ/ M 36cm2
_
6. lrqdf PQRS Pp6f ju{ 5 . h;sf] kl/ldlt
32cm 5 . QR sf] dWolaGb' T xf] eg] UTR sf]
If]qkmn kQf nufpg'xf]; . -pQ/ M 16cm2
_
7. lrqdf PQRS Pp6f ;=r= xf] . RQ nfO{ U ;Dd
nDafOPsf] 5 . olb PQ=20cm / QT=10cm eP
PUS sf] If]qkmn lgsfNg'xf]; . -pQ/ M 100cm2
_
8. lrqdf IL||JK, IL=26cm, IJ=10cm /
IJL=900
eP ILK sf] If]qkmn kQf
nufpg'xf]; . -pQ/ M 120cm2
_
9. lbOPsf] lrqdf, O j[Qsf] s]Gb| laGb', olb
∡OQR = 480 eP ∡QPR sf] dfg slt xf]nf < P O
RQ
D C E
BA
P S T
Q
R
A
B
D
C
28 cm
16 cm
20 cm
A D E
B C
P S
T
RQU
P S U
Q T R
I L
KJ
26 cm
10 cm

5. 5
10. ;“u}sf] lrqdf
/ eP sf] dfg lgsfNg'xf]; .
pQ/ M 1200
11. lrqdf / :kz{ /]vf xf]
eg] sf] gfk lgsfNg'xf]; . pQ/ M
12. lrqdf hLjfx¿ PQ||RS / QTS=200
eP
PTR sf] gfk lgsfNg'xf]; .
-pQ/ M200
_
13. lrqdf O j[Qsf] s]Gb| laGb' xf] .
AB=BC=AC eP BOC sf] gfk kQf
nufpg'xf]; .
-pQ/ M 1200
_
14. O s]Gb| ePsf] j[Qdf PRO=300
eP PQR
sf] gfk kQf nufpg'xf]; .
-pQ/ M 1200
_
15. lrqdf O j[Qsf] s]Gb| POR=1400
/
QPO=200
eP QRO sf] gfk kQf
nufpg'xf]; .
-pQ/ M 500
_
16. O s]Gb| ePsf] j[Qdf PQT=700
eP
ORP sf] gfk kQf nufpg'xf]; .
-pQ/ M 200
_
T
U
S
15o
75o
x
P
RQ
T
B
A
C
N
66o
O
T
SR
P Q
A
B C
O
P R
O
300
Q
Q
S
P R
O
T
700
1400
200
O
P R
Q

6. 6
17. O s]Gb| ePsf] j[Qdf OBA=320
eP
ACB sf] gfk kQf nufpg'xf]; .
-pQ/ M 580
_
18. O s]Gb| ePsf] j[Qdf PSQ=270
eP POQ /
PRQ sf] gfk kQf nufpg'xf]; .
-pQ/ M 540
, 270
)
19. lrqdf O s]Gb| laGb' xf] . PQR = 1200
/
STQ=750
eP STR sf] gfk kQf
nufpg'xf]; . -pQ/ M 450
_
20. lrqdf AB=BC / ACB=220
eP ADC sf]
gfk kQf nufpg'xf]; .
-pQ/ M 440
_
21. lrqdf O s]Gb| AC / BC :kz{ /]vfx¿ x'g .
a[xt sf]0f AOB=2600
eP ACB sf] gfk
kQf nufpg'xf]; .
-pQ/ M 800
_
22. O s]Gb| ePsf] cw{ j[Qdf OR=5 / QR=8 eP
PR sf] gfk kQf nufpg'xf]; .
-pQ/ M 6_
23. O s]Gb| ePsf] j[Qdf =500
/ KOJ=540
eP IOL / IKO sf] gfk kQf nufpg'xf]; .
-pQ/ M 760
/ 270
_
320
O C
BA
270
S
R
O
QP
1200
P Q O T
SR
750
B
220 C
D
A
E
A
B
2600
O
5
R
QOP
8
540
JI
K
L
O

7. 7
24. O s]Gb| ePsf] j[Qdf ++++++OA / BC k/:k/ nDa
¿kdf sfl6Psf 5g . olb COA=800
eP
OAB sf] gfk kQf nufpg'xf]; . -pQ/ M 500
_
25. O s]Gb| ePsf] j[Qdf BC Jof; TAN :kz{ /]vf
/ BCA=660
eP CAN sf] gfk kQf
nufpg'xf]; . -pQ/ M 240
_
26. O s]Gb| ePsf] TAN :kz{ /]vf 5 .
BAN=700
eP AOB sf] gfk kQf
nufpg'xf]; . -pQ/ M 1400
_
27. rj|mLo rt'e'{h PQRS df PS=RS /
PQR=1300
eP PRS sf] gfk kQf
nufpg'xf]; .
-pQ/ M 650
_
-v_ nfdf] pQ/ cfpg] k|Zgx¿ M
pbfx/0f M 1
lbOPsf] lrqdf /
eP sf] If]qkmn / sf] If]qkmn
a/fa/ x'G5g egL k|dfl0ft ug'{xf]; .
;dfwfg M
yfxf lbOPsf] M / 5g .
k|dfl0ft ug'{kg]{ M sf] If]qkmn sf] If]qkmn (1)
B
F
G
A
C
E
D
800
B
AO
C
M
A
C
N
700
O
T
B
660
A
N
C
B
O
T
1300 R
S
P
Q

8. 8
k|df0f M
tYox¿ sf/0fx¿
1. cfwf/ / pxL ;dfgfGt/ /]vfx¿aLr ag]sf
;=r= / lqe'hsf] ;DaGw (1)
2. cfwf/ / pxL ;dfgfGt/ /]vfx¿aLr ag]sf]
;=r= / lqe'hsf] ;DaGw
3. t/ pxL cfwf/ / pxL ;dfgfGt/ /]vfx¿aLr ag]sf
;=r= x¿ ePsf]n] . (1)
4. (1)
k|dfl0ft eof] .
pbfx/0f M 2
lbOPsf] lrqdf, ABCD ;dfgfGt/ rt'e'{h xf] . ;=r= cGtu{t /x]sf] s'g}
laGb' M 5 h;;“u A, B, C, / D qmdzM hf]l8Psf 5g . l;4 ug'{xf]; M
ABM +DCM = ABCD If]qkmndf
;dfwfg M
yfxf lbOPsf] M (i) ABCD ;=r= xf] .
(ii) ;dfgfGt/ rt'e{h leq /x]sf] laGb' M ;“u A, B, C / D qmdzM hf]l8Psf] 5 .
l;4 ug'{ kg]{ M ABM + DCM = ABCD If]qkmndf
h'lSt M AB jf DC sf] ;dfgfGt/ x'g] u/L laGb' M af6 EMF lvrf}“ . (1)
k|df0f M
tYox¿ sf/0fx¿
1.(i) ABEF ;dfgfGt/ rt'e{h xf] . ;Dd'v e'hfx¿ ;dfgfGt/
ePsfn]
(ii) ABM = ABEF Pp6} cfwf/ AB / pxL
;dfgfGt/ /]vfx¿aLr x'gfn] (1)
2. (i) CDFE ;=r= xf] . ;Dd'v e'hfx¿ ;dfgfGt/
ePsfn]
(ii) DCM = CDFE Pp6} cfwf/ DC / pxL
;dfgfGt/ /]vfx¿aLr x'gfn] (1)
3.(i) ABM + DCM = ABEF + CDFE tYo 1 sf] (ii) / 2 sf] (ii) af6
(ii) ABM + DCM = ABCD l;ªuf] 6'qm] tYo (1)
k|dfl0ft eof] .
M
F
E
A D
B C

9. 9
pbfx/0f M 3 ;=r= df n] /
n] nfO{ ;dlåefhg u/]sf 5 .
;=r= x'G5 egL k|dfl0ft ug'{xf]; .
yfxf lbOPsf] M ;=r= xf] . sf] cw{s / sf] cw{s 5g .
k|dfl0ft ug'{kg]{ M ;=r= xf] . (1)
k|df0f M
tYox¿ sf/0fx¿
1. / df
i.
ii.
iii.
iv.
i. ;=r= sf ;Dd'v sf]0fx¿ ePsf]n] .
ii. ;=r= sf ;Dd'v e'hfx¿ ePsf]n] .
iii. a/fa/ sf]0fsf cfwf sf]0fx¿ ePsf]n] .
iv. sf]=e'=sf] tYo cg';f/
(1)
2. 2. lqe'hx¿ cg'¿k ePsf]n]
3. t/ 3. PsfGt/ sf]0fx¿ ePsf]n]
4. 4. tYo (2) / (3) af6
5. 5. ;ªutsf]0fx¿ a/fa/ ePsf]n] (1)
6. ctM Pp6f ;=r=xf] . 6. ;Dd'v e'hfx¿ ;dfGt/ ePsf]n] . (1)
k|dfl0ft eof] .
pbfx/0f M 4
lrqdf eP / x'G5 egL
k|dfl0ft ug'{xf]; .
yfxf lbOPsf] M df 5 .
k|dfl0ft ug'{kg]{ M / (1)
P
S T
Q
R
P U Q
S T R

10. 10
k|df0f M
tYox¿ sf/0fx¿
1.
2.
3.
4.
5.
6.
1. ePsf]n]
2. rj|mLo rt'e'{h ePsf]n]
3. tYo 1 / 2 af6
4. qmdfut leqL sf]0fsf] of]u ePsf]n]
5. a/fa/sf]0fsf ;ªutsf]0fx¿ ePsf]n]
6. a/fa/ sf]0fx¿sf ;Dd'v e'hfx¿ ePsf]n]
(1)
(1)
(1)
k|dfl0ft eof] .
pbfx/0f : 5
lrqdf / 5 . Pp6f
;dfgfGt/ rt'e'{h x'G5 egL k|dfl0ft ug'{xf]; .
yfxf lbOPsf] M ,
k|dfl0ft ug'{kg]{ M Pp6f ;=r= xf] .
k|df0f M
tYox¿ sf/0fx¿
1.
2.
3.
4. Pp6f ;=r= xf] .
1. ePsf]n]
2. rj|mLo rt'e'{hsf] Pp6f e'hf nDAofPsf]n]
3. tYo 1 / 2 af6
4. Ps hf]8f e'hfx¿ ;dfgfGt/ / ;Dd'v sf]0f
a/fa/ ePsf]n]
k|dfl0ft eof] .
pbfx/0f : 6 O s]Gb| ePsf j[Qdf eP j[Qsf
laGb'x¿ x'g egL k|dfl0ft ug'{xf]; .
yfxf lbOPsf] M O s]Gb| ePsf] j[Qdf 5 .
k|dfl0ft ug'{kg]{ M j[Qsf laGb'x¿ . (1)
k|df0f M
QP
T S R
B
A
O
C
X

11. 11
tYox¿ sf/0fx¿
1. / df
i.
ii.
iii.
iv.
v.
1.
i. plx j[Qsf cw{Jof;x¿
ii. yfxf lbOPsf]
iii. ;femf e'hf
iv. e'=e'=e'= tYocg';f/ (1)
2.
3. t/
2. lqe'hx¿ cg'¿k ePsf]n]
3. OA=OC ePsf]n]
4. 4. tYo (2) / (3) af6 (1)
5. AOXB j[Qsf laGb'x¿ x'g . 5. plx rfk OX df ag]sf sf]0fx¿ a/fa/
ePsf]n]
(1)
k|dfl0ft eof] .
pbfx/0f 7 :
;“u}sf] lrqdf O j[Qsf] s]Gb|laGb' xf] . olb OCB=CED eP
EOAB x'G5 egL k|dfl0ft ug'{xf]; .
yfxf lbOPsf] M O s]Gb| ePsf] j[Qdf OCB = CED 5 .
k|dfl0ft ug'{kg]{ M EOAB (1)
tYox¿ sf/0fx¿
1. OCB = CED
2. OCB df, OCB = OBC
3. CED = OBC
4. OAE / ABC df
(i) CED = OBC
(ii) OAE = BAC
(iii) AOE = ACB
(i) t/ ACB=900
(ii) AOE =900
5. OEAB
1. yfxf lbOPsf]
2. OC=OB ePsf]n]
3. tYo 1 / 2 af6
4.
(i) tYo 3 af6
(ii) ;femf sf]0f ePsf]n]
(iii) af“sLsf sf]0fx¿
(iv) cw{j[Qdf ag]sf] kl/lwsf]0f
(v) tYo 4 sf] (iii) / (iv) af6
5. tYo 4 sf] (iv) af6
(1)
(1)
(1)
k|dfl0ft eof] .
BOA
C
D
E

12. 12
lbOPsf pbfx/0fx¿sf] ;xfotfn] lgDglnlvt ;d:ofx¿ ;dfwfg ug'{xf]; M
-v_ nfdf] pQ/ cfpg] k|Zgx¿ M
1. lbOPsf] lrqdf ;dfgfGt/ rt'e'{h ABCD sf]
ljs0f{ BD df /x]sf] s'g} laGb' X ;“u A / C
hf]l8Psf 5g . l;4 ug'{xf]; .
(i) AXB = BXC
(ii) DXC = CDAX.
2. ;=r= sf] e'hf df s'g} laGb' X
5 . BX / AD nfO{ E ;Dd nDAofOPsf] 5 .
C / E hf]8bf / sf] If]qkmn
a/fa/ x'G5g egL k|dfl0ft ug'{xf]; .
3. lbOPsf] lrqdf AE||BC / BE||CD 5 . ABC sf] If]qkmn
= BED x'G5 egL egL k|dfl0ft ug'{xf]; .
4. lbOPsf] lrqdf, PQ = PR 5 . A / B qmdzM
/ sf dWo laGb'x¿ x'g . AB n] PQ /
PR sf] qmdzM M / N df sf6]sf 5g . l;4
ug'{xf]; M MQ = NR.
5. lrqdf O s]Gb| / AC=BD eP CX=DX x'G5
egL k|dfl0ft ug'{xf]; .
6. lbOPsf] lrqdf, O j[Qsf] s]Gb| laGb' / ABCD
rj|mLo rt'e'{h xf] . olb ∡ABD = ∡DAE eP
BCEF klg rj|mLo rt'e'{h x'G5 egL k|dfl0ft
ug'{xf]; .
BA
A BM N
P
Q R
B
X
D
A
C
O
A B
CD E
F
O
D C
A B
X
D C
E
X
A E
B C
D

13. 13
7. O / O' s]Gb| laGb' ePsf b'O{ j[Qx¿sf] :kz{
laGb' X eP x'G5 egL k|dfl0ft
ug'{xf]; .
8. ;“u}sf] lrqdf PTR = QUS eP QPS = PSR /
PQ||RS x'G5 egL k|dfl0ft ug'{xf]; .
9. lrqdf ILNM Pp6f ;=r= xf] . j[Qn] IM sf] J / N sf]
K df sf6]sf] 5 . JKL = JMN x'G5 egL k|dfl0ft
ug'{xf]; .
B
A
O
X
O'
U
Q
SR
P
T
M
NKL
I J