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.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles 

Solution 2

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 3

Given O is the centre of the circle and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAOB = 70o.Calculate the value of:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesOCA ; (ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesOAC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 3

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 6

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 8

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 9

Solution 9

Question 10

In the figure given below, ABCD is a cyclic quadrilateral in which Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAD= 75o ; Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABD= 58o and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADC = 77o. Find:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBDC, (ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD, (iii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCA.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 10

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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 Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCAB = 75o and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCBA = 50o. Find the value of Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDAB + Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABD.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 12

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 13

ABCD is a cyclic quadrilateral in a circle with centre O. IfSelina Solutions Icse Class 10 Mathematics Chapter - CirclesADC = 130o, find Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 13

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 14

In the figure given alongside, AOB is a diameter of the circle and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAOC = 110o, find Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBDC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 14

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 15

In the following figure, O is the centre of the circle;Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAOB = 60o and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBDC = 100o, find Selina Solutions Icse Class 10 Mathematics Chapter - CirclesOBC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 15

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 16

In ABCD is a cyclic quadrilateral in whichSelina Solutions Icse Class 10 Mathematics Chapter - CirclesDAC = 27o,Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDBA = 50o and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADB = 33o. Calculate (i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDBC, (ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDCB, (iii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCAB.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 16

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 17

In the figure given alongside, AB and CD are straight lines through the centre O of a circle. If Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAOC = 80o and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCDE = 40o . Find the number of degrees in: (i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDCE; (ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 17

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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 Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAD = 96o, find Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBFE.

(ii) Prove that AD is parallel to FE.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 19

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 21

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 25

In the figure given RS is a diameter of the circle. NM is parallel to RS and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesMRS = 29o

Calculate : (i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesRNM ; (ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesNRM.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 25

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 26

In the figure given alongside, AB || CD and O is the centre of the circle. If Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADC = 25o; find the angle AEB. Give reasons in support of your answer.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 26

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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, Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAOE = 150o, DAO = 51o. Calculate the sizes of the angles CEB and OCE.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 31

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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 Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAPB = ao. Calculate, in terms of ao, the value of:

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Give reasons for your answers clearly.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 33

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 34

In the given figure, O is the centre of the circle and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC = 55o. Calculate the values of x and y.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 34

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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 Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD = 2Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABE.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 35

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 36

ABCD is a cyclic quadrilateral in which AB is parallel to DC and AB is a diameter of the circle. Given Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBED = 65o; calculate: (i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDAB , (ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBDC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 36

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 37

In the given figure, AB is a diameter of the circle. Chord ED is parallel to AB and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesEAB = 63o; calculate: (i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesEBA , (ii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 37

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAEB = Selina Solutions Icse Class 10 Mathematics Chapter - Circles

(Angle in a semicircle is a right angle)

Therefore Selina Solutions Icse Class 10 Mathematics Chapter - Circles EBA = Selina Solutions Icse Class 10 Mathematics Chapter - Circles - Selina Solutions Icse Class 10 Mathematics Chapter - CirclesEAB = Selina Solutions Icse Class 10 Mathematics Chapter - Circles - Selina Solutions Icse Class 10 Mathematics Chapter - Circles= Selina Solutions Icse Class 10 Mathematics Chapter - Circles


(ii) AB Selina Solutions Icse Class 10 Mathematics Chapter - Circles ED

Therefore Selina Solutions Icse Class 10 Mathematics Chapter - Circles DEB = EBA = Selina Solutions Icse Class 10 Mathematics Chapter - Circles                        (Alternate angles)

Therefore BCDE is a cyclic quadrilateral

Therefore Selina Solutions Icse Class 10 Mathematics Chapter - Circles DEB  Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD = Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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

Therefore Selina Solutions Icse Class 10 Mathematics Chapter - Circles BCD = Selina Solutions Icse Class 10 Mathematics Chapter - Circles - Selina Solutions Icse Class 10 Mathematics Chapter - Circles = Selina Solutions Icse Class 10 Mathematics Chapter - Circles


Question 38

In the given figure, AB is a diameter of the circle with centre O. DO is parallel to CB and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDCB = 120o; calculate:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDAB, (ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDBA, (iii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDBC, (iv) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADC.

Also, show that the Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAOD is an equilateral triangle.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 38

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 39

In the given figure, I is the incentre of the Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC. BI when produced meets the circumcirle of Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC at D. Given Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC = 55° and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesACB = 65o ; calculate: (i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDCA, (ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDAC, (iii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDCI, (iv) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAIC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 39

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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: Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 41

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 42

In the given figure, AB = AC = CD and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADC = 38o. Calculate:

(i) Angle ABC

(ii) Angle BEC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 42

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 44

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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 Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCAB = xo ; find (in terms of x) the values of:

Selina Solutions Icse Class 10 Mathematics Chapter - Circles.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 48

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 49

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

(i) DAB  (ii) DBA

Solution 49
  1. ABCD is a cyclic quadrilateral

mDAB = 180° - DCB

= 180° - 130°

= 50°

  1. In ∆ADB,

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. GivenSelina Solutions Icse Class 10 Mathematics Chapter - CirclesROS = Selina Solutions Icse Class 10 Mathematics Chapter - Circles; calculate Selina Solutions Icse Class 10 Mathematics Chapter - CirclesRTS.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 50

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 51

In the given figure, PQ is a diameter. Chord SR is parallel to PQ. Given that Selina Solutions Icse Class 10 Mathematics Chapter - CirclesPQR = Selina Solutions Icse Class 10 Mathematics Chapter - Circles; calculate

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 51

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 52

AB is the diameter of the circle with centre O. OD is parallel to BC and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAOD = Selina Solutions Icse Class 10 Mathematics Chapter - Circles; calculate the numerical values of:

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 52

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 53

In the given figure, the centre O of the small circle lies on the circumference of the bigger circle. If Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAPB = Selina Solutions Icse Class 10 Mathematics Chapter - Circles and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD = Selina Solutions Icse Class 10 Mathematics Chapter - Circles; find:

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 53

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 54

In the given figure, Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAD = Selina Solutions Icse Class 10 Mathematics Chapter - Circles , Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABD = Selina Solutions Icse Class 10 Mathematics Chapter - Circles and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBDC = Selina Solutions Icse Class 10 Mathematics Chapter - Circles; find:

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Hence, show that AC is a diameter.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 54

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 55

In a cyclic quadrilateral ABCD, Selina Solutions Icse Class 10 Mathematics Chapter - CirclesA :Selina Solutions Icse Class 10 Mathematics Chapter - CirclesC = 3 : 1 and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesB : Selina Solutions Icse Class 10 Mathematics Chapter - CirclesD = 1 : 5; find each angle of the quadrilateral.

Solution 55

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 56

The given figure shows a circle with centre O andSelina Solutions Icse Class 10 Mathematics Chapter - CirclesABP =Selina Solutions Icse Class 10 Mathematics Chapter - Circles. Calculate the measure of

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 56

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 57

In the given figure, M is the centre of the circle. Chords AB and CD are perpendicular to each other. If Selina Solutions Icse Class 10 Mathematics Chapter - CirclesMAD =x and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC = y,

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 57

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 2

In the following figure, AD is the diameter of the circle with centre O. chords AB, BC and CD are equal. If Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDEF = 110Selina Solutions Icse Class 10 Mathematics Chapter - Circles, calculate :

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAFE,(ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesFAB.

Solution 2

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles


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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 4

The given figure show a circle with centre O. also, PQ = QR = RS and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesPTS = 75°. Calculate:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesPOS,

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesQOR,

(iii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesPQR.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 4

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAOB,

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesACB,

(iii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 5

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 6

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

Solution 6

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 7

In the given figure. AB = BC = CD and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC = 132° calculate:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAEB,

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAED,

(iii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCOD.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 7

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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 :

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCAB,

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADB.

Solution 8

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 9

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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:

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

(i) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADC,

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAD,

(iii) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC,

(iv) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAEC.

Solution 10

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Chapter 17 - Circles Exercise Ex. 17(C)

Question 1

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 1

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 2

In the given figure, ABC is a triangle in which Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC = 30Selina Solutions Icse Class 10 Mathematics Chapter - Circles. Show that BC is equal to the radius of the circum-circle of the triangle ABC, whose centre is O.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 2

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 3

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

Solution 3


Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Selina Solutions Icse Class 10 Mathematics Chapter - Circles


Question 4

In the given figure, chord ED is parallel to diameter AC of the circle. Given Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCBE =Selina Solutions Icse Class 10 Mathematics Chapter - Circles, calculateSelina Solutions Icse Class 10 Mathematics Chapter - CirclesDEC.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 4

Selina Solutions Icse Class 10 Mathematics Chapter - Circles


 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles


Question 5

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

Solution 5

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Selina Solutions Icse Class 10 Mathematics Chapter - Circles









Question 6

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

(i) BDC

(ii) BEC

(iii) BAC

Selina Solutions Icse Class 10 Mathematics Chapter - Circles 

Solution 6

Selina Solutions Icse Class 10 Mathematics Chapter - Circles 

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles





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 Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADC =Selina Solutions Icse Class 10 Mathematics Chapter - Circles,Selina Solutions Icse Class 10 Mathematics Chapter - CirclesFAE =Selina Solutions Icse Class 10 Mathematics Chapter - Circles; determineSelina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD. Given reason in support of your answer.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 8

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Question 9

If I is the incentre of triangle ABC and AI when produced meets the cicrumcircle of triangle ABC in points D. ifSelina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC =Selina Solutions Icse Class 10 Mathematics Chapter - Circlesand Selina Solutions Icse Class 10 Mathematics Chapter - Circles= Selina Solutions Icse Class 10 Mathematics Chapter - Circles.calculate:

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

(i)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDBC (ii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesIBC (iii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBIC.

Solution 9

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Question 10

In the given figure, AB = AD = DC = PB and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesDBC = xo. Determine, in terms of x :

(i)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABD, (ii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesAPB.

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 10

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles




Question 11

In the given figure; ABC, AEQ and CEP are straight lines. Show thatSelina Solutions Icse Class 10 Mathematics Chapter - CirclesAPE andSelina Solutions Icse Class 10 Mathematics Chapter - CirclesCQE are supplementary.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 11

Selina Solutions Icse Class 10 Mathematics Chapter - Circles




Selina Solutions Icse Class 10 Mathematics Chapter - Circles


Question 12

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

IfSelina Solutions Icse Class 10 Mathematics Chapter - CirclesADC = Selina Solutions Icse Class 10 Mathematics Chapter - Circles, find angle BOC.

Solution 12

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Selina Solutions Icse Class 10 Mathematics Chapter - Circles




Question 13

In a cyclic-quadrilateral PQRS, angle PQR =Selina Solutions Icse Class 10 Mathematics Chapter - Circles. Sides SP and RQ prouduced meet at point A: whereas sides PQ and SR produced meet at point B.

IfSelina Solutions Icse Class 10 Mathematics Chapter - CirclesA : Selina Solutions Icse Class 10 Mathematics Chapter - CirclesB =2 : 1 ; find angles A and B.

Solution 13

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Question 14

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles


 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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.

IfSelina Solutions Icse Class 10 Mathematics Chapter - CirclesDCF : Selina Solutions Icse Class 10 Mathematics Chapter - CirclesF : Selina Solutions Icse Class 10 Mathematics Chapter - CirclesE = 3 : 5 : 4, find the angles of the cyclic quadrilateral ABCD.

Solution 15

Selina Solutions Icse Class 10 Mathematics Chapter - Circles




Selina Solutions Icse Class 10 Mathematics Chapter - Circles


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.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 16

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



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 ofSelina Solutions Icse Class 10 Mathematics Chapter - CirclesCAD.

Further if the lenghof arc AC is twice the lengthof arc BC find : (a)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC (b) Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 17

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 18

In cyclic quadrilateral ABCD; AD = BC,Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC=Selina Solutions Icse Class 10 Mathematics Chapter - Circles andSelina Solutions Icse Class 10 Mathematics Chapter - CirclesCBD= Selina Solutions Icse Class 10 Mathematics Chapter - Circles; find ;

(i)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD (ii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCA

(iii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC (iv)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADC

Solution 18

Selina Solutions Icse Class 10 Mathematics Chapter - Circles



Selina Solutions Icse Class 10 Mathematics Chapter - Circles


Question 19

In the given figure, Selina Solutions Icse Class 10 Mathematics Chapter - CirclesACE = Selina Solutions Icse Class 10 Mathematics Chapter - Circles and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCAF=Selina Solutions Icse Class 10 Mathematics Chapter - Circles; find the values of a, b and c.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 19

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 20

In the given figure, AB is parallel to DC ,Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCE = Selina Solutions Icse Class 10 Mathematics Chapter - Circlesand Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC = Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Find

(i)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCAD (ii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCBD (iii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADC

Solution 20

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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 =Selina Solutions Icse Class 10 Mathematics Chapter - Circles.

Solution 21

Selina Solutions Icse Class 10 Mathematics Chapter - Circles


Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 22

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

Show that DP bisects angle ADB.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 22

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Question 23

In the figure, given below , AD = BC, Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBAC = Selina Solutions Icse Class 10 Mathematics Chapter - Circles and Selina Solutions Icse Class 10 Mathematics Chapter - CirclesCBD = Selina Solutions Icse Class 10 Mathematics Chapter - Circlesfind:

(i)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCD

(ii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBCA

(iii)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesABC

(iv)Selina Solutions Icse Class 10 Mathematics Chapter - CirclesADB

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

Solution 23

Selina Solutions Icse Class 10 Mathematics Chapter - Circles


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°.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles 

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.

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

 

Selina Solutions Icse Class 10 Mathematics Chapter - Circles

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

Selina Solutions Icse Class 10 Mathematics Chapter - Circles 

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 = Selina Solutions Icse Class 10 Mathematics Chapter - CirclesBOD…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°