ML Aggarwal Solution Class 10 Chapter 15 Circles Exercise 15.2

Exercise 15.2

Question 1

If O is the centre of the circle, find the value of x in each of the following figures (using the given information):





















Sol :

From the figure

(i) ABCD is a cyclic quadrilateral

From the figure

Ext. DCE=BAD
BAD=x

Now arc BD subtends BOD at the centre and BAD at the remaining part of the circle.

BOD=2BAD=2x

2x=150x=75


(ii) BCD+DCE=180

(Linear pair)

Figure to be added

BCD+80=180

BCD=18080=100

∵Arc BAD subtends reflex ∠BOD at the centre and ∠BCD at the remaining part of the circle

Reflex BOD=2BCD

x0=2×100=200


(iii) In ΔACB

Figure to be added

CAB+ABC+ACB=180

(Angles of a triangle)

But ACB=90 (Angle in a semicircle)

25+90+ABC=180

115+ABC=180

ABC=180115=65

ABCD is a cyclic quadrilateral

ABC+ADC=180

(Opposite angles of a cyclic quadrilateral)

⇒65°+x°=180°

⇒x°=180°-65°=115°


Question 2

(a) In the figure (i) given below, O is the centre of the circle. If ∠AOC = 150°, find (i) ∠ABC (ii) ∠ADC.










(b) In the figure (i) given below, AC is a diameter of the given circle and ∠BCD = 75°. Calculate the size of (i) ∠ABC (ii) ∠EAF.











Sol :

(a) Given, ∠AOC = 150° and AD = CD

We know that an angle subtends by an arc of a circle

at the centre is twice the angle subtended by the same arc

at any point on the remaining part of the circle.


(i) AOC=2×ABC

ABC=AOC2=1502=75


(ii) From figure,ABCD is a cyclic quadrilateral

ABC+ADC=180

Sum of opposite angles in a cyclic quadrilateral is 180

75+ADC=180

ADC=18075

ADC=105


(b) (i)AC is the diameter of the circle.

ABC=90 (Angle in a semi circle)

BAD+75=180

(BCD=75)

BAD=18075=105

But EAF=BAD

(Vertically opposite angles)

EAF=105


Question 3

(a) In the figure, (i) given below, if ∠DBC = 58° and BD is a diameter of the circle, calculate:

(i) ∠BDC (ii) ∠BEC (iii) ∠BAC












(b) In the figure (if) given below, AB is parallel to DC, ∠BCE = 80° and ∠BAC = 25°. Find:
(i) ∠CAD (ii) ∠CBD (iii) ∠ADC (2008)





Sol :

(a) ∠DBC = 58°
BD is diameter
∠DCB = 90° (Angle in semi circle)
(i) In ∆BDC
∠BDC + ∠DCB + ∠CBD = 180°
∠BDC = 180°- 90° – 58° = 32°












(ii) BEC=18032=148

(opp. angle of cyclic quadrilateral)

(iii) BAC=BDC=32

(Angles in same segment)


(b) In the figure, ABDC

BCE=80 and BAC=25

ABCD is a cyclic quadrilateral and DC is produced to E






(i) ∴Ext. ∠BCE= Interior ∠A
80=BAC+CAD
80=25+CAD
CAD=8025=55


(ii) But CAD=CBD

∴∠CBD=55°


(iii) ∠BAC=∠BDC

(Angles in the same segment)

∴∠BDC=25°  (∵∠BAC=25°)

Now AB||DC and BD is the transversal

∴∠BDC=∠ABD (Alternate angles)

∠ABD=25°

∴∠ABC=∠ABD+∠CBD

=25°+55°=80°

But ∠ABC+∠ADC=180°

(Opposite angles of a cyclic quadrilateral)

⇒80°+∠ADC=180°

⇒∠ADC=180°-80°=100°


Question 4

(a) In the figure given below, ABCD is a cyclic quadrilateral. If ∠ADC = 80° and ∠ACD = 52°, find the values of ∠ABC and ∠CBD.











(b) In the figure given below, O is the centre of the circle. ∠AOE =150°, ∠DAO = 51°. Calculate the sizes of ∠BEC and ∠EBC.











Sol :

(a) In the given figure, ABCD is a cyclic quadrilateral

∠ADC = 80° and ∠ACD = 52°

To find the measure of ∠ABC and ∠CBD


ABCD is a cyclic quadrilateral 
ABC+ADC=180
(sum of opposite angles =180)

ABC+80=180

ABC=18080=100

In ΔADC,

∠DAC=180°-∠ADC+∠ACD

=180°-(80°+52°)

=180°-132°=48°

But CBD=DAC

(Angles in the same segment)

CBD=48


(b) In the given figure, O is the centre of the circle.






AOE=150,DAO=51

To find BEC and EBC

ABED is a cyclic quadrilateral

Ext. BEC=DAB=51

AOE=150


Ref. AOE=360150=210

Now arc ABE subtends AOE at the centre

and ADE at the remaining part of the circle.

ADE=12 Ref. AOE=12×210=105

But Ext. EBC=ADE=105

Hence BEC=51 and EBC=105


Question 5

(a) In the figure (i) given below, ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. Given that ∠BEF = 80°, find ∠ABC.

(b) In the figure (ii) given below, ABCD is a cyclic trapezium in which AD is parallel to BC and ∠B = 70°, find:

(i)∠BAD (ii) DBCD.










Sol :
(a) ADFE is a cyclic quadrilateral
Ext. ∠FEB = ∠ADF
⇒ ∠ADF = 80°
ABCD is a parallelogram
∠B = ∠D = ∠ADF = 80°
or ∠ABC = 80°
(b)In trapezium ABCD, AD || BC
(i) ∠B + ∠A = 180°
⇒ 70° + ∠A = 180°
⇒ ∠A = 180° – 70° = 110°
∠BAD = 110°
(ii) ABCD is a cyclic quadrilateral
∠A + ∠C = 180°
⇒ 110° + ∠C = 180°
⇒ ∠C = 180° – 110° = 70°
∠BCD = 70°

Question 6

(a) In the figure given below, O is the centre of the circle. If ∠BAD = 30°, find the values of p, q and r.










(b) In the figure given below, two circles intersect at points P and Q. If ∠A = 80° and ∠D = 84°, calculate

(i) ∠QBC (ii) ∠BCP










Sol :
(a) (i) ABCD is a cyclic quadrilateral

A+C=180

30+p=180

p=18030=150


(ii) Arc BD subtends BOD at the centre and BAD at the remaining part of the circle

BOD=2BAD

q=2×30=60

BAD and BED are in the same segment of the circle

BAD=BED30=r

r=30


(b) Join PQ

AQPD is a cyclic quadrilateral.

A+QPD=180

80+QPD=180

QPD=18080=100

and D+AQP=180

84+AQP=180

AQP=18084=96

Now PQBC is a cyclic quadrilateral

Ext. QPD=QBC

QBC=100

and ext. AQP=BCP

BCP=96


Question 7

(a) In the figure given below, PQ is a diameter. Chord SR is parallel to PQ . Given ∠PQR = 58°, calculate (i) ∠RPQ (ii) ∠STP

(T is a point on the minor arc SP)









(b) In the figure given below, if ∠ACE = 43° and ∠CAF = 62°, find the values of a, b and c (2007)












Sol :

(a) In ∆PQR,
∠PRQ = 90° (Angle in a semi circle) and ∠PQR = 58°
∠RPQ = 90° – ∠PQR = 90° – 58° = 32°
SR || PQ (given)

SRP=RPQ=32 (alternate angles)

Now PRST is a cyclic quadrilateral,
STP+SRP=180
STP+32=180
STP=18032=148


(b) In the given figure,
ACE=43 and CAF=62
Now, in AEC
ACE+CAE+AEC=180
43+62+AEC=180
105+AEC=180
AEC=180105=75
But ABD+AED=180

(Sum of opposite angles of a cyclic quadrilateral)

and AED=AEC
a+75=180
a=18075=105
(Angles in the alternate segment)

c=62
Now in Δ BAF
a+62+b=180
105+62+b=180
167+b=180
b=180167=13

Hence a=105°
, b=13°
, c=62°

Question 8

(a) In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB.
(b) In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x : y : z = 3 : 4 : 5, find the values of x, y and z.



















Sol :

(a) Construction: Join BC, and AC then
ABCD is a cyclic quadrilateral.

In quadrilateral ABCD
ABC+ADC=180

( Sum of opposite s of a cyclic quadrilateral)

ABC+120=180ABC=60

Now in ABC
ABC+ACB+CAB=180
(InΔABC,ACB=90)
(angle in a semicircle)

CAB+60+90=180
CAB=30


(b) ∠DCF=∠BCE=x
(Vertically opposite angle)










Now In ΔDCF
Ext. 2=x+z ....(i)
and in ΔCBE
Ext. 1=x+y ...(ii)
Adding (i) and (ii)

x+y+x+z=1+2
2x+y+z=180
(\because ABCD is a cyclic quad.) ...(iii)

But x : y : z = 3 : 4 : 5

xy=34

y=43x

and xz=35

z=53x

Substituting the value of y and z in (iii)

2x+43x+53x=180
6x+4x+5x=180×3
15x=180×3
x=180×315=36
y=43x=36×43=48
and z=53x=36×53=60

Hence , x=36°, y=48° and z=60°


Question 9

(a) In the figure (i) given below, ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E. If ∠ADE = 70° and ∠OBA = 45°, calculate
(i) ∠OCA (ii) ∠BAC
(b) In figure (ii) given below, ABF is a straight line and BE || DC. If ∠DAB = 92° and ∠EBF = 20°, find :
(i) ∠BCD (ii) ∠ADC.









Sol :

(a) ABCD is a cyclic quadrilateral
Figure to be added

Given, ABCD is a quadrilateral inscribed in a circle with centre O. Also, CD is a chord produced to E

ADE=70 and OBA=45
ADE+ADC=180 (Linear pair)
ADC=18070=110
Again, D+B=180

(Opposite angles of a cyclic quadrilateral are supplementary)

110+B=180
B=70

OBA+OBC=70
45+OBC=70
OBC=7045
OBC=25

Also, AOC=140

(Angle subtended by an arc of a circle at the centre is double than the angle drawn subtended by it at any point on the remaining part of a circle)

Now in ΔOAC
OAC+OCA+AOC=180
( Sum of all s of a tiangle)

OAC+OCA+140=180
OAC+OAC+140=180

OA= OC, radii of same circle implies that angle opposite to equal sides are equal)

2OAC=40OAC=20

Similarly in ΔOAB

OAB=OBA=45

( OA=OB, radii of same circle )

Now BAC=OAB+OAC
BAC=45+20
BAC=65


(b) ABCD is a cyclic quadrilateral

BCD+DAB=180









BCD+92=180
BCD=18092=88.

Again Ext. CBF=ADC

ADC=CBF=CBE+EBF
=BCD+EBF

(DCBE and alternate angles are equal)
=88+20=108

Question 10

(a) In the figure (ii) given below, PQRS is a cyclic quadrilateral in which PQ = QR and RS is produced to T. If ∠QPR = 52°, calculate ∠PST.










(b) In the figure (ii) given below, O is the centre of the circle. If ∠OAD = 50°, find the values of x and y.












Sol :
(a) PQRS is a cyclic quadrilateral in which
PQ = QR











in ΔPQR

QPR=QRP=52

PQR=180(QPR+QRP)

=180(52+52)=180104=76

Now in cyclic quad. PQRS,
Ext. PST=PQR=76




(b) ABCD is a cyclic quadrilateral
OAD+BCD=180

50+x=180
x=18050=130












In ΔOAD,OA=OD
(Radii of the same circle)

ODA=OAD=50
Now Ext. BOD=ODA+OAD
y=50+50=100


Question 11

(a) In the figure (i) given below, O is the centre of the circle. If ∠COD = 40° and ∠CBE = 100°, then find :
(i) ∠ADC
(ii) ∠DAC
(iii) ∠ODA
(iv) ∠OCA.
(b) In the figure (ii) given below, O is the centre of the circle. If ∠BAD = 75° and BC = CD, find :
(i) ∠BOD
(ii) ∠BCD
(iii) ∠BOC
(iv) ∠OBD (2009)











Sol :
(a) (i) ∴ ABCD is a cyclic quadrilateral.
∴ Ext. ∠CBE = ∠ADC
⇒ ∠ADC = 100°

(ii) Arc CD subtends ∠COD at the centre
and ∠CAD at the remaining part of the circle
∴ ∠COD = 2 ∠CAD
CAD=12COD=12×40=20









(iii) In ΔCOD, OC=OD
(radii of the same circle)

CDO=DCO

But CDO+DCO+COD=180
CDO+CDO+COD=180
2CDO+40=180
2CDO=18040=140

CDO=1402=70

Now ODA=ADCCDO
=10070=30


(iv) OCA=OCDACD
=70{180ADCCAD}
=70{18010020}
=70{180120}=7060=10


(b) Given BAD=75, chord BC= chord CD
Construction : Join OC and BD











(i) BOD=2×BAD
BOD=2×75=150

(ii) In quadrilateral ABCD,
 BCD+BAD=180
(Sum of opposite s in quadrilateral)
BCD=18075
BCD=105

(iii) BOC=12BOD
BOC=12×150=75

(iv) In ΔOBD
OBD+ODC+BOD=180
(Sum of all s of a triangle is 180 )
OBD+ODC+150=180
OBD+OBD+150=180
(BC=CD)

2OBD=30
OBD=15


Question 12

In the given figure, O is the centre and AOE is the diameter of the semicircle ABCDE. If AB = BC and ∠AEC = 50°, find :
(i) ∠CBE
(ii) ∠CDE
(iii) ∠AOB.
Prove that OB is parallel to EC.







Sol :
In the given figure,
O is the centre of the semi-circle ABCDE
and AOE is the diameter. AB = BC, ∠AEC = 50°

To prove : OB||EC and find
(i) ∠CBE,
(ii) ∠CDE and ∠AOB

Proof : AECB is a cyclic quadrilateral
∴∠AEC+∠ABC=180°

50+ABE+EBC=180
50+90+EBC=180
140+EBC=180
(∵∠ABE is in a semi-circle)

EBC=180140=40
But BEDC is a cyclic quadrilateral

EBC+CDE=180
40+CDE=180

CDE=18040=140
AB=BC

AEB=BEC=12AEC

=12×50=25

In Δ OBE
OB=OE (radii of the same circle)

OBE=OEB=25
and Ext. AOB=OBE+OEB
=25+25=50
AOB=50
AOB=OEC (each 50°)

But these are corresponding angles

OBEC

Question 13

(a) In the figure (i) given below, ED and BC are two parallel chords of the circle and ABE, ACD are two st. lines. Prove that AED is an isosceles triangle.









(b) In the figure (ii) given below, SP is the bisector of ∠RPT and PQRS is a cyclic quadrilateral. Prove that SQ = RS.











Sol :
(a) Given: Chord BC || ED,
ABE and ACD are straight lines.
To Prove: ∆AED is an isosceles triangle.
Proof: BCDE is a cyclic quadrilateral.
Ext. ∠ABC = ∠D …(i)
But BC || ED (given)


ABC=E

(corresponding angle) ...(ii)

From (i) and (ii)
D=E

In ΔAED,
D=E (proved)

AE=AD
(sides opposite to equal angles)

Hence, ΔAED is an isosceles triangle Q.E.D



(b) Given : In a circle , PQRS is a cyclic quadrilateral . QP is produced to T sucht that PS is the bisector of ∠RPT

To prove : SQ=RS

Proof : ∵PQRS is a cyclic quadrilateral

∴Ext.∠TPS=∠SRQ...(i)
and ∠RPS=∠RQS..(ii)
(Angles in the same segment)

But ∠TPS=∠RPS
(∵PS is the bisector of ∠TPR)

∴from (i) and (ii)
∠SRQ=∠RQS

Now in ΔSQR,
∠SRQ=∠RQS proved

∴ SQ=RS 
(sides opposite to equal angles)

Q.E.D

To prove : SQ=RS
Proof : ∵PQRS is a cyclic quadrilateral
∴ Ext. ∠TPS=∠SRQ ...(i)
and ∠RPS=∠RQS...(ii)
(∵ PS is the bisector of ∠TPR)
∴ from (i) and (ii)
∠SRQ=∠RQS

Now in ΔSQR,
∠SRQ=∠RQS (proved)

∴ SQ=RS
(sides opposite to equal angles)

Q.E.D

Question 14

In the given figure, ABC is an isosceles triangle in which AB = AC and circle passing through B and C intersects sides AB and AC at points D and E. Prove that DE || BC.














Sol :

In the given figure,
∆ABC is an isosceles triangle in which AB = AC.
A circle passing through B and C intersects
sides AB and AC at D and E.
To prove: DE || BC
Construction : Join DE.
∵ AB = AC
∠B = ∠C (angles opposite to equal sides)
But BCED is a cyclic quadrilateral
Ext. ∠ADE = ∠C
= ∠B (∵ ∠C = ∠B)
But these are corresponding angles
DE || BC
Hence proved.

Question 15

(a) Prove that a cyclic parallelogram is a rectangle.
(b) Prove that a cyclic rhombus is a square.
Sol :
(a) ABCD is a cyclic parallelogram.









To prove: ABCD is a rectangle
Proof : ABCD is a parallelogram
∠A = ∠C and ∠B = ∠D
(opposite angles are equal)

∴ABCD is a cyclic quadrilateral
A+C=180
A+A=180  (A=C)

2A=180
A=1802=90

C=A=90

Similarly, we can prove that
B=D=90
ABCD is a rectangle


(b) Given : ABCD is a cyclic rhombus
To prove : ABCD is a square
Proof : ∵ABCD is a rhombus
∴∠A=∠C and ∠B=∠D
(opposite angles are equal)

But ABCD is a cyclic quadrilateral
A+C=180 and B+D=180
A+A=180 (A=C)

2 A=180

A=1802=90


C=A=90

Similarly we can prove that
B=D=90
ABCD is a square.

Question 16

In the given figure, chords AB and CD of the circle are produced to meet at O. Prove that triangles ODB and OAC are similar. Given that CD = 2 cm, DO = 6 cm and BO = 3 cm, area of quad. CABD
calculate AB. Also find  area of quad. CABD  area of Δ OAC 










Sol :
In the given figure, AB and CD are chords of a circle.
They are produced to meet at O.
To prove : (i) ∆ODB ~ ∆OAC
If CD = 2 cm, DO = 6 cm, and BO = 3 cm
To find : AB and also area of the

ABCD area  of ΔOAC

Construction : Join AC and BD








Proof:
(i) InΔODB and ΔOAC
Ext. ODB=A
(Ext. angle of a cyclic quadrilateral is equal to its interior opposite angle) Similarly, OBD=C

O=O (common)
Δ ODB Δ OAC (AAA axiom)


(ii) ODOA=OBOC
6OA=36+2

6OA=38

OA=6×83=16

AB=OAOB=163=13 cm


(iii) ΔOACΔODB

 area ΔOAC area ΔODB=OC2OB2=8232=649...(i)

Subtracting 1 from both sides, we get

 area ΔOAC area ΔODB1=6491

 area ΔOAC area ΔODB area ΔODB=6499

 area quadrilateral CABD  area ΔOBD=559

and  area quadrilateral CABD  area ΔOAC=5564 [From (i)]

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