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

Chapter Exercises

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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14

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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8

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

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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.

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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.

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11

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

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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.

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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

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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.

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Multiple Choice Questions (MCQ)

1

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

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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

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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 = ?

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4

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

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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

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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

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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

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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

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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

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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

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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

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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

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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 = ?

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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

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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 = ?

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16

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

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17

In the given figure, the length of BC is

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18

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

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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 = ?

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20

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

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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

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22

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

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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

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24

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

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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

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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

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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

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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

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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

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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 =?

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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

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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 = ?

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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

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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

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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 = ?

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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

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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

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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 = ?

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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

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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

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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

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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

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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 = ?

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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

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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

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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

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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


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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

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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

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50

Which of the following statements is not true?

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51

Which of the following statements is not true?

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52

Which of the following statements is not true?

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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:


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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


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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:


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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 = ?

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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

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3

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

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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

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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.


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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.

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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.

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8

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

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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.

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10

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

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11

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

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12

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

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13

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

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14

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

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15

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

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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.

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17

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

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18

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

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19

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the center.

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20

PQ is a chord of length 16 cm of a circle of radius 10 cm. The tangents at P and Q intersect at a point T as shown in the figure.

Find the length of TP.


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