# Solution of Chapter 10. Circles (RD Sharma - Mathematics Book)

## Exercise 10.1

1

Fill in the blanks:

(i) The common point of a tangent and the circle is called………… .

(ii) A circle may have ………… parallel tangents.

(iii) A tangent to a circle intersects it in ……… points(s).

(iv) A line intersecting a circle in two points is called a ………… .

(v) The angle between tangent at a point on a circle and the radius through the point is ………… .

2

How many tangents can a circle have?

3

O is the centre of a circle of radius 8 cm. The tangent at a point A on the circle cuts a line through O at B such that AB = 15 cm. Find OB.

4

If the tangent at a point P to a circle with centre O cuts a line through O at Q such that PQ = 24 cm and OQ = 25 cm. Find the radius of the circle.

## Exercise 10.2

1

If PT is a tangent at T to a circle whose centre is O and OP = 17 cm, OT = 8 cm, find the length of the tangent segment PT.

2

Find the length of a tangent drawn to a circle with radius 5 cm, from a point 13 cm from the centre of the circle.

3

A point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 10 cm. Find the radius of the circle.

4

If from any point on the common chord of two intersecting circles, tangents be drawn to the circles, prove that they are equal.

5

If the sides of a quadrilateral touch a circle, prove that the sum of a pair of opposite sides is equal to the sum of the other pair.

6

If AB, AC, PQ are tangents in Fig. 10.51 and AB = 5 cm, find the perimeter of . 7

Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

8

In Fig. 10.52, PQ is tangent at a point R of the circle with centre O. If , find . 9

If PA and PB are tangents from an outside point P. such that PA = 10 cm and . Find the length of chord AB.

10

From an external point P, tangents PA are drawn to a circle with centre O. If CD is the tangent to the circle at a point E and PA = 14 cm, find the perimeter of .

11

In Fig. 10.53, ABC is a right triangle right-angled at B such that BC = 6 cm and AB = 8 cm. Find the radius of its incircle. 12

From a point P, two tangents PA and PB are drawn to a circle with centre O. If OP = diameter of the circle, show that is equilateral.

13

Two tangent segments PA and PB are drawn to a circle with centre O such that . Prove that OP = 2 AP.

14

If is isosceles with AB = AC and C (O, r) is the incircle of the touching BC at L, prove that L bisect BC.

15

In Fig. 10.54, a circle touches all the four sides of a quadrilateral ABCD with AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD. 16

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

17

In fig. 10.55, O is the centre of the circle and BCD is tangent to it at C. Prove that . 18

Two circles touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR 19

In Fig 10.57, a circle is inscribed in a quadrilateral ABCD in which . If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius r of the circle. 20

In Fig. 10.58, there are two concentric circles with centre O of radii 5 cm and 3 cm. from an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP. 21

In Fig. 10.59, AB is a cord of length 16 cm of a circle of radius 10 cm. The tangents at A and B intersect at a point P. Find the length of PA. 22

In Fig. 10.60, PA and PB are tangents from an external point P to a circle with centre O. LN touches the circle at A. Prove that PL + LM = PN + MN. 23

In Fig. 10.61, BDC is a tangent to the given circle at point D such that BD = 30 cm and CD = 7 cm. The other tangents BE and CF are drawn respectively from B and C to the circle and meet when produced at A making BAC a right angle triangle. Calculate (i) AF (ii) radius of the circle. 24

In Fig. 10.62, . The tangents to the circle at P and Q intersect at a point T. Prove that PQ and OT are right bisectors of each other. 25

In Fig. 10.63, two tangents AB and AC are drawn to a circle with centre O such that . Prove that OA = 2AB. 26

In Fig. 10.64, BC is a tangent to the circle O. OE bisects AP. Prove that . 27

The lengths of three consecutive sides of a quadrilateral circumscribing a circle are 4 cm, 5 cm and 7 cm respectively. Determine the length of the fourth side.

28

In Fig. 10.65, common tangents PQ and RS to two circles intersect at A. Prove that PQ = RS. 29

Equal circles with centres O and O’ touch each other at X. OO’ produced to meet a circle with centre O’, at A. AC is tangent to the circle whose centre is O. O’ D is perpendicular to AC. Find the value of . 30

In Fig. 10.67, OQ:PQ=3:4 and perimeter of = 60 cm. Determine PQ, OR and OP. 31

Two concentric circles are of diameters 30 cm and 18 cm. Find the length of the chord of the larger circle which touches the smaller circle.

32

A triangle PQR is drawn to circumscribe a circle of radius 8 cm such that the segments QT and TR, into which QR is divided by the point of contact T, are of lengths 14 cm and 16 cm respectively. If area of is 336 cm2, find the sides PQ and PR.

33

In Fig. 10.68, a is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 8 cm and 6 cm respectively. Find the lengths of sides AB and AC, when area of is 84 cm2. 34

In Fig. 10.69, AB is a diameter of a circle with centre O and AT is a tangent. If , find . 35

In Fig. 10.70, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that ∠RPQ = 30o. A chord RS is drawn parallel to the tangent PQ. Find  ∠RQS. ## CCE - Formative Assessment

1

In Fig. 10.72, PA and PB are tangents to the circle drawn from an external point P. CD are a third tangent touching the circle at Q. If PB = 10 cm and CQ = 2 cm, what is the length PC? 2

What is the distance between two parallel tangents of a circle of radius 4 cm?

3

The length of tangent from a point A at a distance of 5 cm from the center of the circle is 4 cm. What is the radius of the circle?

4

Two tangents TP and TQ are drawn from an external point T to a circle with center O as shown in Fig. 10.73. If they are inclined to each other at an angle of 100°, then what is the value of POQ? 5

What the distance between two parallel tangents to a circle of radius 5 cm?

6

In Q. No. 1, if PB = 10 cm, what is the perimeter of Δ PCD? 7

In Fig. 10.74, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm and BC = 7 cm, then find the length of BR. 8

In Fig. 10.75, Δ ABC is circumscribing a circle. Find the length of BC. 9

In Fig. 10.76, CP and CQ are tangents from an external point C to a circle with centre O. AB is another tangent which touches the circle at R. If CP = 11 cm and BR = 4 cm, find the length of BC. [Hint: We have, CP = 11 cm

Now, BR = BQ [Tangents drawn from B) BQ = 4 cm BC = CQ - BQ = (11 - 4)cm = 7 cm

10

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

11

In Fig. 10.77, PA and PB are tangents to the circle with centre O such that APB = 50°. Write the measure of OAB 1

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q such that OQ = 12 cm. Length PQ is cm

2

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is

3

The length of the tangent from a point A at a circle, of radius 3 cm, is 4 cm. The distance of A from the centre of the circle is

4

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80° then POA is equal to

5

If TP and TQ are two tangents to a circle with centre O so that POQ = 110°, then, PTQ is equal to

6

PQ is a tangent to a circle with centre 0 at the point P. If A Δ OPQ is an isosceles triangle, then OQP is equal to

7

Two equal circles touch each other externally at C and AB is a common tangent to the circles. Then, ACB =

8

ABC is a right angled triangle, right angled at B such that BC = 6 cm and AB = 8 cm. A circle with centre O is inscribed in ΔABC. The radius of the circle is

9

PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that POR = 120° , then OPQ is

10

If four sides of a quadrilateral ABCD are tangential to a circle, then

11

The length of the tangent drawn from a point 8 cm away from the centre of a circle of radius 6 cm is

12

AB and CD are two common tangents to circles which touch each other at C. If D lies on AB such that CD = 4 cm, then AB is equal to

13

In Fig. 10.78, if AD, AE and BC are tangents to the circle at D, E and F respectively. Then, 14

In Fig. 10.79, RQ is a tangent to the circle with centre O. If SQ = 6 cm and QR = 4 cm, then OR = 15

In Fig. 10.80, the perimeter of ΔABC is 16

In Fig. 10.81, AP is a tangent to the circle with centre O such that OP = 4 cm and OPA = 30°. Then, AP = 17

AP and PQ are tangents drawn from a point A to a circle with centre O and radius 9 cm. If OA = 15 cm, then AP + AQ =

18

At one end of a diameter PQ of a circle of radius 5 cm, tangent XPY is drawn to the circle. The length of chord AB parallel to XY and at a distance of 8 cm from P is

19

If PT is tangent drawn from a point P to a circle touching it at T and O is the centre of the circle, then OPT + POT =

20

In the adjacent figure, if AB = 12 cm, BC = 8 cm and AC = 10 cm, then AD = 21

In Fig. 10.83, if AP = PB, then 22

In Fig. 10.84, if AP = 10 cm, then BP = 23

In Fig. 10.85, if PR is tangent to the circle at P and Q is the centre of the circle, then POQ = 24

In Fig. 10.86, if quadrilateral PQRS circumscribes a circle, then PD + QB = 25

In Fig. 10.87, two equal circles touch each other at T, if QP = 4.5 cm, then QR = 26

In Fig. 10.88, APB is a tangent to a circle with centre O at point P. If QPB = 500, then the measure of POQ is 27

In Fig. 10.89, if tangents PA and PB are drawn to a circle such that APS = 30° and chord AC is drawn parallel to the tangent PB, then ABC = 28

In Fig. 10.90, PR = 29

Two circles of same radii r and centres O and O' touch each other at P as shown in Fig. 10.91. If 00' is produced to meet the circle C (O', r) at A and AT is a tangent to the circle C(O, r) such that O'Q AT. Then AO: AO' = 30

Two concentric circles of radii 3 cm and 5 cm are given. Then length of chord BC which touches the inner circle at P is equal to 31

In Fig. 10.93, there are two concentric circles with centre O. PR and PQS are tangents to the inner circle from point plying on the outer circle. If PR = 7.5 cm, then PS is equal to 32

In Fig. 10.94, if AB = 8 cm and PE = 3 cm, then AE = 33

In Fig. 10.95, PQ and PR are tangents drawn from P to a circle with centre O. If OPQ = 35°, then 34

In Fig. 10.96, if TP and TQ are tangents drawn from an external point T to a circle with centre O such that TQP = 60°, then OPQ = 35

In Fig. 10.97, the sides AB, BC and CA of triangle ABC, touch a circle at P, Q and R respectively. If PA = 4 cm, BP = 3 cm and AC = 11cm, then length of BC is 36

In Fig. 10.98, a circle touches the side DF of AEDF at H and touches ED and EF produced at K and M respectively. If EK = 9 cm, then the perimeter of ΔEDF is 37

In Fig. 10.99, DE and DF are tangents from an external point D to a circle with centre A. If DE = 5 cm and DE DF, then the radius of the circle is

38

In Fig. 10.100, a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches sides BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD = 23 cm, B = 90° and DS = 5 cm, then the radius of the circle (in cm) is 39

In a right triangle ABC, right angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is

40

Two circles touch each other externally at P. AB is a common tangent to the circle touching them at A and B. The value of APB is

41

In Fig. 10.101, PQ and PR are two tangents to a circle with centre O. If QPR = 46°, then QOR equals 42

In Fig. 10.102, QR is a common tangent to the given circles touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is  