# Solution of Chapter 12. Circles (RS Aggarwal - Mathematics Book)

## Exercise 12A

1

Find the length of tangent drawn to a circle with radius 8 cm from a point 17 cm away from the center of the circle.

2

A point P is 25 cm away from the center of a circle and the length of tangent drawn from P to the circle is 24 cm. Find the radius of the circle.

3

Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the larger circle which touches the smaller circle.

4

In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, find the lengths of AD, BE and CF.

5

In the given figure, PA and PB are the tangent segments to a circle with center 0. Show that the points A, O, B and P are concyclic.

6

In the given figure, the chord AB of the larger of the two concentric circles, with center O, touches the smaller circle at C. Prove that AC = CB.

7

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

8

A circle is inscribed in a LABC touching AB, BC and AC at P, Q and R respectively. If AB = 10 cm, AR = 7 cm and CR = 5 cm, find the length of BC.

9

In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.

10

In the given figure, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC.

11

In the given figure, O is the center of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to the outer and inner circle respectively. If PA = 10 cm, find the length of PB up to one place of decimal.

12

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 6 cm and 9 cm respectively. If the area of ABC = 54 cm2 then find the lengths of sides AB and AC.

13

PQ is a chord of length 4.8 cm of a circle of radius 3 cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of TP.

14

Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its center.

15

In the given figure, a circle with center O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD = 23 cm, LB = 90° and DS = 5 cm then find the radius of the circle.

16

In the given figure, O is the center of the circle and TP is the tangent to the circle from an external point T. If PBT = 30°, prove that BA : AT = 2 : 1.

## Exercise 12B

1

In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of side AD.

2

In the given figure, PA and PB are two tangents to the circle with center O. If APB = 50° then what is the measure of OAB.

3

In the given figure, O is the center of a circle. PT and PQ are tangents to the circle from an external point P. If TPQ = 70°, find TRQ.

4

In the given figure, common tangents AB and CD to the two circles with centers O1 and O2 intersect at E. Prove that AB = CD.

5

If PT is a tangent to a circle with center O and PQ is a chord of the circle such that QPT = 70°, then find the measure of POQ.

6

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4 cm and 3 cm respectively. If the area of ΔABC = 21 cm2 then find the lengths of sides AB and AC.

7

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

8

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

9

In the given figure, two tangents RQ and RP are drawn from an external point R to the circle with center 0. If PRQ = 120°, then prove that OR = PR + RQ.

10

In the given figure, a circle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. If AB = 14 cm, BC = 8 cm and CA = 12 cm. Find the lengths AD, BE and CF.

11

In the given figure, 0 is the center of the circle. PA and PB are tangents. Show that AOBP is a cyclic quadrilateral.

12

In two concentric circles, a chord of length 8 cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5 cm then find the radius of the smaller circle.

13

In the given figure, PQ is a chord of a circle with center 0 and PT is a tangent. If QPT = 60°, find P

14

In the given figure, PA and PB are two tangents to the circle with center O. If APB = 60° then find the measure of OAB.

## Multiple Choice Questions (MCQ)

1

The number of tangents that can be drawn from an external point to a circle is

2

In the given figure, RQ is a tangent to the circle with center O. If SQ = 6 cm and QR = 4 cm, then OR is equal to

3

In a circle of radius 7 cm, tangent PT is drawn from a point P such that PT = 24 cm. If O is the center of the circle, then length OP = ?

4

Which of the following pairs of lines in a circle cannot be parallel?

5

The chord of a circle of radius 10 cm subtends a right angle at its center. The length of the chord (in cm) is

6

In the given figure, PT is a tangent to the circle with center O. If OT = 6 cm and OP = 10 cm, then the length of tangent PT is

7

In the given figure, point P is 26 cm away from the center 0 of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then, the radius of the circle is

8

PQ is a tangent to a circle with center O at the point P. If ΔOPQ is an isosceles triangle, then OQP is equal to

9

In the given figure, AB and AC are tangents to the circle with center O such that BAC = 40°. Then, BOC is equal to

10

If a chord AB subtends an angle of 60° at the center of a circle, then the angle between the tangents to the circle drawn from A and B is

11

In the given figure, O is the center of two concentric circles of radii 6 cm and 10 cm. AB is a chord of outer circle which touches the inner circle. The length of chord AB is

12

In the given figure, AB and AC are tangents to a circle with center 0 and radius 8 cm. If OA = 17 cm, then the length of AC (in cm) is

13

In the given figure, 0 is the center of a circle, AOC is its diameter such that ACB = 50°. If AT is the tangent to the circle at the point A then BAT = ?

14

In the given figure, O is the center of a circle, PQ is a chord and PT is the tangent at P. If POQ = 70°, then TPQ is equal to

15

In the given figure, AT is a tangent to the circle with center O such that OT = 4 cm and OTA = 30°. Then, AT = ?

16

If PA and PB are two tangents to a circle with center O such that AOB = 110° then APB is equal to

17

In the given figure, the length of BC is

18

In the given figure, if AOD = 135° then BOC is equal to

19

In the given figure, 0 is the center of a circle and PT is the tangent to the circle. If PQ is a chord such that QPT = 50° then POQ = ?

20

In the given figure, PA and PB are two tangents to the circle with center O. If APB = 60° then OAB is

21

If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm then the length of each tangent is

22

In the given figure, PQ and PR are tangents to a circle with center A. If QPA = 27° then QAR equals

23

In the given figure, PA and PB are two tangents drawn from an external point P to a circle with center C and radius 4 cm. If PA PB, then the length of each tangent is

24

If PA and PB are two tangents to a circle with center O such that APB = 80°. Then, AOP = ?

25

In the given figure, O is the center of the circle. AB is the tangent to the circle at the point P. If APQ = 58° then the measure of PQB is

26

In the given figure, O is the center of the circle. AB is the tangent to the circle at the point P. If PAO = 30° then CPB + ACP is equal to

27

In the given figure, PQ is a tangent to a circle with center O. A is the point of contact. If PAB = 67°, then the measure of AQB is

28

In the given figure, two circles touch each other at C and AB is a tangent to both the circles. The measure of ACB is

29

O is the center of a circle of radius 5 cm. At a distance of 13 cm from O, a point P is taken. From this point, two tangents PQ and PR are drawn to P the circle. Then, the area of quad. PQOR is

30

In the given figure, PQR is a tangent to the circle at Q, whose center is O and AB is a chord parallel to PR such that BQR = 70°. Then, AQB =?

31

The length of the tangent from an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from the center of the circle is

32

In the given figure, 0 is the center of a circle, BOA is its diameter and the tangent at the point P meets BA extended at T. If PBO = 30° then PTA = ?

33

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

34

To draw a pair of tangents to a circle, which is inclined to each other at an angle of 45°, we have to draw tangents at the end points of those two radii, the angle between which is

35

In the given figure, O is the center of a circle; PQL and PRM are the tangents at the points Q and R respectively and S is a point on the circle such that SQL = 50° and SRM = 60°. Then, QSR = ?

36

In the given figure, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T, are of lengths 12 cm and 9 cm respectively. If the area of ΔPQR = 189 cm2 then the length of side PQ is

37

In the given figure, 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 is

38

In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S. If AP = 5 cm, BC = 7 cm and CS = 3 cm. Then, the length AB = ?

39

In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S. If AP = 6 cm, BP = 5 cm, CQ = 3 cm and DR = 4 cm then perimeter of quad. ABCD is

40

In the given figure, O is the center of a circle, AB is a chord and AT is the tangent at A. If AOB = 100° then BAT is equal to

41

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 is

42

In the given figure, a circle is inscribed in a quadrilateral ABCD touching its sides AB, BC, CD and AD at P, Q, R and S respectively. lithe radius of the circle is 10 cm, BC = 38 cm, PB = 27 cm and AD CD then the length of CD is

43

In the given figure, LABC is right - angled at B such that BC = 6 cm and AB = 8 cm. A circle with center O has been inscribed inside the triangle. OP AB, OQ BC and OR AC. If OP = OQ = OR = x cm then x = ?

44

Quadrilateral ABCD is circumscribed to a circle. If AB = 6 cm, BC = 7 cm and CD = 4 cm then the length of AD is

45

In the given figure, PA and PB are tangents to the given circle such that PA = 5 cm and APB = 60°. The length of chord AB is

46

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

47

In the given figure, three circles with centers A, B, C respectively touch each other externally.

If AB = 5 cm, BC = 7 cm and CA = 6 cm then the radius of the circle with center A is

48

In the given figure, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC = 4 cm then the length of AP is

49

In the given figure, O is the center of two concentric circles of radii 5 cm and 3 cm. From an external point P tangents PA and PB are drawn to these circles. If PA = 12 cm then PB is equal to

50

Which of the following statements is not true?

51

Which of the following statements is not true?

52

Which of the following statements is not true?

53

Assertion - and - Reason Type

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

54

Assertion - and - Reason Type

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code

55

Assertion - and - Reason Type

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

## Formative Assessment (Unit Test)

1

In the given figure, O is the center of a circle, PQ is a chord and the tangent PT at P makes an angle of 50° with PQ. Then, POQ = ?

2

If the angle between two radii of a circle is 130° then the angle between the tangents at the ends of the radii is

3

If tangents PA and PB from a point P to a circle with center O are drawn so that APB = 80° then POA = ?

4

In the given figure, AD and AE are the tangents to a circle with center O and BC touches the circle at F. If AE = 5 cm then perimeter of ΔABC is

5

In the given figure, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively.

If AB = x cm, BC = 7 cm, CR = 3 cm and

AS = 5 cm, find x.

6

In the given figure, PA and PB are the tangents to a circle with center O. Show that the points A, 0, B, P are concyclic.

7

In the given figure, PA and PB are two tangents from an external point P to a circle with center O. If PBA = 65°, find OAB and APB.

8

Two tangent segments BC and BD are drawn to a circle with center O such that CBD = 120°. Prove that OB = 2BC.

9

Fill in the blanks.

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

(ii) A circle can have ………..parallel tangents at the most.

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

(iv) A circle can have ………..tangents.

10

Prove that the lengths of two tangents drawn from an external point to a circle are equal.

11

Prove that the tangents drawn at the ends of the diameter of a circle are parallel.

12

In the given figure, if AB = AC, prove that BE = CE.

13

If two tangents are drawn to a circle from an external point, show that they subtend equal angles at the center.

14

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

15

Prove that the parallelogram circumscribing a circle, is a rhombus.

16

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

17

A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal.

18

Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.