# SELINA Solutions for Class 10 Maths Chapter 17 - Circles

Get seamless access to accurate Selina Solutions for ICSE Class 10 Mathematics Chapter 17 Circles at TopperLearning. Revisit the important topics of circles by solving problems related to angle properties and cyclic properties. Also, understand how to provide proof that a given line segment is a diameter of a circle.

Study the ICSE Class 10 Maths Selina solutions to be capable of analysing the given data for performing the required calculations. If you feel like attending the lecture on circles again, watch our online video lessons. To revisit the concepts, our revision notes and Frank solutions may also benefit you.

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## Chapter 17 - Circles Exercise Ex. 17(A)

Question 1

In the given figure, O is the centre of the circle. respectively. Find angle AOC Show your steps of working.

Solution 1

Question 2

In the given figure, BAD = 65°, ABD = 70°, BDC = 45°

(i) Prove that AC is a diameter of the circle.

(ii) Find ACB.

Solution 2

Question 3

Given O is the centre of the circle and AOB = 70o.Calculate the value of:

(i) OCA ; (ii) OAC.

Solution 3

Question 4

In each of the following figures, O is the centre of the circle. Find the values of a, b and c.

Solution 4

Question 5

In each of the following figures, O is the centre of the circle. Find the values of a, b, c and d.

Solution 5

Question 6

In the figure, AB is common chord of the two circles. If AC and AD are diameters; prove that D, B and C are in a straight line. O1 and O2 are the centres of two circles.

Solution 6

Question 7

In the figure, given below, find:

Show steps of your working.

Solution 7

Question 8

In the figure, given below, O is the centre of the circle. If

Solution 8

Question 9

Solution 9

Question 10

In the figure given below, ABCD is a cyclic quadrilateral in which BAD= 75o ; ABD= 58o and ADC = 77o. Find:

(i) BDC, (ii) BCD, (iii) BCA.

Solution 10

Question 11

In the figure given below, O is the centre of the circle and triangle ABC is equilateral. Find:

(i) ADB, (ii) AEB.

Solution 11

Question 12

Given CAB = 75o and CBA = 50o. Find the value of DAB + ABD.

Solution 12

Question 13

ABCD is a cyclic quadrilateral in a circle with centre O. IfADC = 130o, find BAC.

Solution 13

Question 14

In the figure given alongside, AOB is a diameter of the circle and AOC = 110o, find BDC.

Solution 14

Question 15

In the following figure, O is the centre of the circle;AOB = 60o and BDC = 100o, find OBC.

Solution 15

Question 16

In ABCD is a cyclic quadrilateral in whichDAC = 27o,DBA = 50o and ADB = 33o. Calculate (i) DBC, (ii) DCB, (iii)CAB.

Solution 16

Question 17

In the figure given alongside, AB and CD are straight lines through the centre O of a circle. If AOC = 80o and CDE = 40o . Find the number of degrees in: (i) DCE; (ii) ABC.

Solution 17

Question 18

In the figure given below, AC is a diameter of a circle, whose centre is O. A circle is described on AO as diameter. AE, a chord of the larger circle, intersects the smaller circle at B. Prove that AB = BE.

Solution 18

Question 19

In the following figure,

(i) if BAD = 96o, find BCD and BFE.

(ii) Prove that AD is parallel to FE.

Solution 19

Question 20

Prove that:

(i) the parallelogram, inscribed in a circle, is a rectangle.

(ii) the rhombus, inscribed in a circle, is a square.

Solution 20

Question 21

In the following figure, AB = AC. Prove that DECB is an isosceles trapezium.

Solution 21

Question 22

Two circles intersect at P and Q. Through P diameters PA and PB of the two circles are drawn. Show that the points A, Q and B are collinear.

Solution 22

Question 23

The figure given below, shows a circle with centre O. Given: AOC = a and ABC = b.

(i) Find the relationship between a and b

(ii) Find the measure of angle OAB, if OABC is a parallelogram.

Solution 23

Question 24

Two chords AB and CD intersect at P inside the circle. Prove that the sum of the angles subtended by the arcs AC and BD as the centre O is equal to twice the angle APC

Solution 24

Question 25

In the figure given RS is a diameter of the circle. NM is parallel to RS and MRS = 29o

Calculate : (i) RNM ; (ii) NRM.

Solution 25

Question 26

In the figure given alongside, AB || CD and O is the centre of the circle. If ADC = 25o; find the angle AEB. Give reasons in support of your answer.

Solution 26

Question 27

Two circles intersect at P and Q. Through P, a straight line APB is drawn to meet the circles in A and B. Through Q, a straight line is drawn to meet the circles at C and D. Prove that AC is parallel to BD.

Solution 27

Question 28

ABCD is a cyclic quadrilateral in which AB and DC on being produced, meet at P such that PA = PD. Prove that AD is parallel to BC.

Solution 28

Question 29

AB is a diameter of the circle APBR as shown in the figure. APQ and RBQ are straight lines. Find:

Solution 29

Question 30

In the given figure, SP is the bisector of angle RPT and PQRS is a cyclic quadrilateral. Prove that: SQ = SR.

Solution 30

Question 31

In the figure, O is the centre of the circle, AOE = 150o, DAO = 51o. Calculate the sizes of the angles CEB and OCE.

Solution 31

Question 32

In the figure, P and Q are the centres of two circles intersecting at B and C. ACD is a straight line. Calculate the numerical value of x.

Solution 32

Question 33

The figure shows two circles which intersect at A and B. The centre of the smaller circle is O and lies on the circumference of the larger circle. Given that APB = ao. Calculate, in terms of ao, the value of:

Solution 33

Question 34

In the given figure, O is the centre of the circle and ABC = 55o. Calculate the values of x and y.

Solution 34

Question 35

In the given figure, A is the centre of the circle, ABCD is a parallelogram and CDE is a straight line. Prove that BCD = 2ABE.

Solution 35

Question 36

ABCD is a cyclic quadrilateral in which AB is parallel to DC and AB is a diameter of the circle. Given BED = 65o; calculate: (i) DAB , (ii) BDC.

Solution 36

Question 37

In the given figure, AB is a diameter of the circle. Chord ED is parallel to AB and EAB = 63o; calculate: (i) EBA , (ii)BCD.

Solution 37

(i) AEB =

(Angle in a semicircle is a right angle)

Therefore EBA = - EAB = - =

(ii) AB ED

Therefore DEB = EBA =                        (Alternate angles)

Therefore BCDE is a cyclic quadrilateral

Therefore DEB  BCD =

[Pair of opposite angles in a cyclic quadrilateral are supplementary]

Therefore BCD =  - =

Question 38

In the given figure, AB is a diameter of the circle with centre O. DO is parallel to CB and DCB = 120o; calculate:

(i) DAB, (ii) DBA, (iii) DBC, (iv) ADC.

Also, show that the AOD is an equilateral triangle.

Solution 38

Question 39

In the given figure, I is the incentre of the ABC. BI when produced meets the circumcirle of ABC at D. Given BAC = 55° and ACB = 65o ; calculate: (i) DCA, (ii) DAC, (iii) DCI, (iv) AIC.

Solution 39

Question 40

A triangle ABC is inscribed in a circle. The bisectors of angles BAC, ABC and ACB meet the circumcircle of the triangle at points P, Q and R respectively. Prove that:

Solution 40

Question 41

Calculate the angles x, y and z if:

Solution 41

Question 42

In the given figure, AB = AC = CD and ADC = 38o. Calculate:

(i) Angle ABC

(ii) Angle BEC.

Solution 42

Question 43

In the given figure, AC is the diameter of circle, centre O. Chord BD is perpendicular to AC. Write down the angles p, q and r in terms of x.

Solution 43

Question 44

In the given figure, AC is the diameter of circle, centre O. CD and BE are parallel. Angle AOB = 80o and angle ACE = 10o. Calculate:

(i) Angle BEC ; (ii) Angle BCD ; (iii) Angle CED.

Solution 44

Question 45

In the given figure, AE is the diameter of circle. Write down the numerical value of . Give reasons for your answer.

Solution 45

Question 46

In the given figure, AOC is a diameter and AC is parallel to ED. If , calculate .

Solution 46

Question 47

Use the given figure to find

.

Solution 47

Question 48

In the given figure, AOB is a diameter and DC is parallel to AB. If CAB = xo ; find (in terms of x) the values of:

.

Solution 48

Question 49

In the given figure, AB is the diameter of a circle with centre O. BCD = 130°. Find:

(i) DAB  (ii) DBA

Solution 49
1. ABCD is a cyclic quadrilateral

mDAB = 180° - DCB

= 180° - 130°

= 50°

mDAB + mADB + mDBA = 180°

50° + 90° + mDBA = 180°

mDBA = 40°

Question 50

In the given figure, PQ is the diameter of the circle whose centre is O. GivenROS = ; calculate RTS.

Solution 50

Question 51

In the given figure, PQ is a diameter. Chord SR is parallel to PQ. Given that PQR = ; calculate

Solution 51

Question 52

AB is the diameter of the circle with centre O. OD is parallel to BC and AOD = ; calculate the numerical values of:

Solution 52

Question 53

In the given figure, the centre O of the small circle lies on the circumference of the bigger circle. If APB = and BCD = ; find:

Solution 53

Question 54

In the given figure, BAD = , ABD = and BDC = ; find:

Hence, show that AC is a diameter.

Solution 54

Question 55

In a cyclic quadrilateral ABCD, A :C = 3 : 1 and B : D = 1 : 5; find each angle of the quadrilateral.

Solution 55

Question 56

The given figure shows a circle with centre O andABP =. Calculate the measure of

Solution 56

Question 57

In the given figure, M is the centre of the circle. Chords AB and CD are perpendicular to each other. If MAD =x and BAC = y,

Solution 57

## Chapter 17 - Circles Exercise Ex. 17(B)

Question 1

In a cyclic-trapezium, the non-parallel sides are equal and the diagonals are also equal.

Prove it.

Solution 1

Question 2

In the following figure, AD is the diameter of the circle with centre O. chords AB, BC and CD are equal. If DEF = 110, calculate :

(i) AFE,(ii) FAB.

Solution 2

Question 3

If two sides of a cycli-quadrilateral are parallel; prove thet:

(i) its other two side are equal.

(ii) its diagonals are equal.

Solution 3

Question 4

The given figure show a circle with centre O. also, PQ = QR = RS and PTS = 75°. Calculate:

(i) POS,

(ii) QOR,

(iii) PQR.

Solution 4

Question 5

In the given figure, AB is a side of a regular six-sided polygon and AC is a side of a regular eight-sided polygon inscribed in the circle with centre O. calculate the sizes of :

(i) AOB,

(ii) ACB,

(iii) ABC.

Solution 5

Question 6

In a regular pentagon ABCDE, inscribed in a circle; find ratio between angle EDA and angel ADC.

Solution 6

Question 7

In the given figure. AB = BC = CD and ABC = 132° calculate:

(i) AEB,

(ii) AED,

(iii) COD.

Solution 7

Question 8

In the figure, O is the centre of the circle and the length of arc AB is twice the length of arc BC. If angle AOB = 108° find :

(i) CAB,

Solution 8

Question 9

The figure shows a circle with centre O. AB is the side of regular pentagon and AC is the side of regular hexagon. Find the angles of triangle ABC.

Solution 9

Question 10

In the given figire, BD is a side of a regularhexagon, DC is a side of a regular pentagon and AD is adiameter. Calculate:

(iii) ABC,

(iv) AEC.

Solution 10

## Chapter 17 - Circles Exercise Ex. 17(C)

Question 1

In the given circle with diametre AB, find the valuv of x.

Solution 1

Question 2

In the given figure, ABC is a triangle in which BAC = 30. Show that BC is equal to the radius of the circum-circle of the triangle ABC, whose centre is O.

Solution 2

Question 3

Prove that the circle drawn on any one a the equalside of an isoscele triangle as diameter bisects the base.

Solution 3

Question 4

In the given figure, chord ED is parallel to diameter AC of the circle. Given CBE =, calculateDEC.

Solution 4

Question 5

The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. Prove it.

Solution 5

Question 6

In the figure, DBC = 58°. BD is a diameter of the circle. Calculate :

(i) BDC

(ii) BEC

(iii) BAC

Solution 6

Question 7

D and E are points on equal sides AB and AC of an isosceles triangle ABC such that AD = AE. Provet that the points B, C, E and D are concyclic.

Solution 7

Question 8

In the given rigure, ABCD is a cyclic eqadrilateral. AF is drawn parallel to CB and DA is produced to point E. If ADC =,FAE =; determineBCD. Given reason in support of your answer.

Solution 8

Question 9

If I is the incentre of triangle ABC and AI when produced meets the cicrumcircle of triangle ABC in points D. ifBAC =and = .calculate:

(i)DBC (ii)IBC (iii)BIC.

Solution 9

Question 10

In the given figure, AB = AD = DC = PB and DBC = xo. Determine, in terms of x :

(i)ABD, (ii)APB.

Hence or otherwise, prove thet AP is parallel to DB.

Solution 10

Question 11

In the given figure; ABC, AEQ and CEP are straight lines. Show thatAPE andCQE are supplementary.

Solution 11

Question 12

In the given, AB is the diameter of the circle with centre O.

IfADC = , find angle BOC.

Solution 12

Question 13

In a cyclic-quadrilateral PQRS, angle PQR =. Sides SP and RQ prouduced meet at point A: whereas sides PQ and SR produced meet at point B.

IfA : B =2 : 1 ; find angles A and B.

Solution 13

Question 14

In the following figure, ABCD is a cyclic quadrilateral in which AD is parallel to BC.

If the bisector of angle A meet BC at point E and the given circle at point F, prove that:

(i) EF = FC (ii) BF = DF

Solution 14

Question 15

ABCD is a cyclic quadrilateral. Sides AB and DC produced meet at point e; whereas sides BC and AD produced meet at point F.

IfDCF : F : E = 3 : 5 : 4, find the angles of the cyclic quadrilateral ABCD.

Solution 15

Question 16

The following figure shows a cicrcle with PR as its diameter. If PQ = 7 cm and QR = 3RS = 6 cm, Find the perimetre of the cyclic quadrilateral PQRS.

Solution 16

Question 17

In the following figure, AB is the diameter of a circle with centre O. If chord AC = chord AD ,prove that:

(i) arc BC = arc DB

(ii) AB is bisector ofCAD.

Further if the lenghof arc AC is twice the lengthof arc BC find : (a)BAC (b) ABC

Solution 17

Question 18

In cyclic quadrilateral ABCD; AD = BC,BAC= andCBD= ; find ;

(i)BCD (ii)BCA

Solution 18

Question 19

In the given figure, ACE = and CAF=; find the values of a, b and c.

Solution 19

Question 20

In the given figure, AB is parallel to DC ,BCE = and BAC =

Find

Solution 20

Question 21

ABCD is a cyclic quadrilalteral of a circle with centre O such that AB is a diameter of this circle and the length of the chord CD is equal to the radius of the circle,.if AD and BC produced meet at P, show that APB =.

Solution 21

Question 22

In the figure, given alongside, CP bisects angle ACB.

Show that DP bisects angle ADB.

Solution 22

Question 23

In the figure, given below , AD = BC, BAC = and CBD = find:

(i)BCD

(ii)BCA

(iii)ABC

Solution 23

Question 24

In the given figure, AD is a diameter. O is the centre of the circle. AD is parallel to BC and CBD = 32°.

Find:

i. OBD

ii. AOB

iii. BED

Solution 24

i. AD is parallel to BC, i.e., OD is parallel to BC and BD is transversal.

Question 25

In the figure given, O is the centre of the circle. DAE = 70°. Find giving suitable reasons, the measure of

i. BCD

ii. BOD

iii. OBD

Solution 25

DAE and DAB are linear pair

So,

DAE + DAB = 180°

DAB = 110°

Also,

BCD + DAB = 180°……Opp. Angles of cyclic quadrilateral BADC

BCD = 70°

BCD = BOD…angles subtended by an arc on the center and on the circle

BOD = 140°

In ΔBOD,

OB = OD……radii of same circle

So,

OBD =ODB……isosceles triangle theorem

OBD + ODB + BOD = 180°……sum of angles of triangle

2OBD = 40°

OBD = 20°