RS Aggarwal - Mathematics

Book: RS Aggarwal - Mathematics

Chapter: 12. Circles

Subject: Maths - Class 10th

Q. No. 35 of Multiple Choice Questions (MCQ)

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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 = ?

As PL and PM are tangents to given circle,


We have,


OR ⏊ PM and OQ ⏊ PL


[Tangents drawn at a point on circle is perpendicular to the radius through point of contact]


So, ∠ORM = ∠OQL = 90°


∠ORM = ∠ORS + ∠SRM


90° = ∠ORS + 60°


∠ORS = 30°


And ∠OQL = ∠OQS + ∠SQL


90° = ∠OQS + 50°


∠OQS = 40°


Now, In △SOR


OS = OQ [radii of same circle]


∠ORS = ∠OSR


[Angles opposite to equal sides are equal]


∠OSR = 30°


[as ∠ORS = 30°]


Now, In △SOR


OS = SQ [radii of same circle]


∠OQS = ∠OSQ


[Angles opposite to equal sides are equal]


∠OSQ = 40° [as ∠OQS = 40°]


As,


∠QSR = ∠OSR + ∠OSQ


∠QSR = 30° + 40° = 70°

Chapter Exercises

Exercise 12A
Exercise 12B
Multiple Choice Questions (MCQ)
Formative Assessment (Unit Test)

More Exercise Questions

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

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24

If PA and PB are two tangents to a circle with center O such that ∠APB = 80°. Then, ∠AOP = ?
25

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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
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

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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
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

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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 = ?
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

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45

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46

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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
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

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51

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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:

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