Circles for ICSE and CBSE | 25 Conceptual Problems

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Chapter Test Circles for CBSE, ICSE | Class X

👨‍🏫 Author: Singh

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Geometry - Circles

This quiz provides a thorough evaluation of key concepts and advanced skills in Circle Geometry, tailored for both CBSE and ICSE curriculums. It covers all core theorems and properties, including: Angle properties (Angle at the centre, Angles in the same segment). Cyclic Quadrilateral properties. Intersecting chord Theorem. Tangent-Secant theorems (Alternate Segment Theorem, Tangent-Radius properties). Perfect for exam preparation, this practice test is designed to challenge your understanding and refine your problem-solving techniques. CIRCLES | CBSE | ICSE . 25 multiple-choice questions designed to test both theoretical understanding and practical application.

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📝 Instructions

• This quiz contains 25 multiple choice questions.
• Select only one correct answer per question.
• Use the navigator to jump between questions.
• Submit when you are finished to see results.

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Question 1

From a point \(P\), two tangents \(PA\) and \(PB\) are drawn to a circle with centre at \(O\) and radius \(r\). If \(OP=2r\), then \(\triangle APB\) is:

Aan isosceles triangle Ba right angle triangle Can equilateral triangle Da scalene triangle

Question 2

In the given figure, \(AB\) is a chord of length \(16\) cm of a circle of radius \(10\) cm. The tangents at \(A\) and \(B\) interesct at point \(P\). The length of \(PA\) is:

A\(\frac{37}{7} \quad cm\) B\(\frac{37}{3} \quad cm\) C\(\frac{40}{7} \quad cm\) D\(\frac{40}{3} \quad cm\)

Question 3

The angle subtended by a tangent \(MN\) at the centre \(O\) of a circle, intercepted between two parallel tangets \(AM\) and \(BN\) of the same circle is:

A\(75^{\circ}\) B\(60^{\circ}\) C\(45^{\circ}\) D\(90^{\circ}\)

Question 4

In the figure \(AB\) is a common tangent of two circles intersecting at \(C\) and \(D\). The value of \(\angle ACB+ \angle ADB\) is:

A\(180^{\circ}\) B\(220^{\circ}\) C\(160^{\circ}\) D\(280^{\circ}\)

Question 5

In the given figure \(O\) is the centre of the circle. \(AB\) and \(AC\) are tangents drawn to the circle from point \(A\). If \(\angle BAC = 65^{\circ}\), then the measure of \(\angle BOC \) is:

A\(100^{\circ}\) B\(135^{\circ}\) C\(135^{\circ}\) D\(115^{\circ}\)

Question 6

\(AB\) is the diameter of the circle with centre \(O\). From a point \(P\) on the circle a perpendicular \(PN\) is drawn on \(AB\). Which of the following is true?

A\(PB^2 = AN.BN\) B\(PB^2 = AB.BN\) C\(PN^2 = AB.BN\) D\(PN^2 = AB.AP\)

Question 7

From an external point \(P\) of a circle with centre \(O\), two tangents \(PS\) and \(PT\) are drawn. \(QS\) is a chord of the circle parallel to \(PT\). If \( \angle SPT = 80^{\circ}\), then the value of \(\angle QST \) is

A\(55^{\circ}\) B\(45^{\circ}\) C\(50^{\circ}\) D\(60^{\circ}\)

Question 8

In the figure given below, \(PA\) is a tangent to the circle with centre \(O\) and \(PCB\) is a straight line. The measure on \(\angle OBC \) is

A\(20^{\circ}\) B\(30^{\circ}\) C\(40^{\circ}\) D\(35^{\circ}\)

Question 9

In the figure given below, a circle with centre \(O\) inscribed inside triangle \(LMN\). \(A\) and \(B\) are the points of tangency. The measure on \(\angle ANB \) is

A\(70^{\circ}\) B\(80^{\circ}\) C\(60^{\circ}\) D\(50^{\circ}\)

Question 10

In the diagram given below, \(\angle EDC = 90^{\circ}\). The tangent drawn to the circle at \(C\) makes an angle of \(50^{\circ}\) with \(AB\) produced.The measure on \(\angle ACB \) is

A\(60^{\circ}\) B\(50^{\circ}\) C\(30^{\circ}\) D\(40^{\circ}\)

Question 11

In the figure given below, \(CD\) is the diameter of the circle which meets the chord \(AB\) at \(P\) such that \(AP = BP = 12 \quad cm\). If \(DP = 8\) cm, find the radius of the circle is

A\(17 \) cm B\(11\) cm C\(7\) cm D\(13\) cm

Question 12

\(ABCD\) is a cyclic quadrilateral. If \(AD=AB\), \(\angle DAC = 60^{\circ}\), \(\angle BDC=50^{\circ}\) then the measure of \(\angle ACD\) is

A\(25^{\circ}\) B\(20^{\circ}\) C\(35^{\circ}\) D\(45^{\circ}\)

Question 13

In the figure given below, two circles touch each other at point \(A\). \(PQ\) is a direct common tangent, the point of contacts being \(P\) and \(Q\) respectively. The measure of \(\angle PAQ\) is

A\(90^{\circ}\) B\(45^{\circ}\) C\(120^{\circ}\) DCannot be determined

Question 14

In the diagram given below, \(O\) is centre of circle. The tangent \(PT\) meets the diameter \(RQ\) produced at \(P\). If \(PT = 6\) cm, \(QR = 9 \) cm, the length of \(PQ\) is ( Hint \(\triangle PQT \sim \triangle PTR\))

A\(1.5\) cm B\(2\) cm C\(3\) cm D\(3.5 \) cm

Question 15

In the given figure AC is the diameter of the circle with centre \(O\). \(CD\) is parallel to \(BE\). If \(\angle AOB = 80^{\circ}\) and \(\angle ACE = 20^{\circ}\), the sum of the angles \(\angle BEC\), \(\angle BCD\) and \(\angle CED\) is

A\(280^{\circ}\) B\(160^{\circ}\) C\(120^{\circ}\) D\(180^{\circ}\)

Question 16

In the given figure, \(O\) is the centre of the circle and \(AB\) is a tangent to the circle at \(B\).If \(\angle PQB =55^{\circ}\), sum of \(x,y\) and \(z\) is

A\(160^{\circ}\) B\(140^{\circ}\) C\(180^{\circ}\) D\(120^{\circ}\)

Question 17

In the figure given below, \(O\) is the centre of the circle and \(SP\) is a tangent.If \(\angle SRT = 65^{\circ}\), find the sum of \(x, y\) and \(z\).

A\(120^{\circ}\) B\(115^{\circ}\) C\(135^{\circ}\) D\(150^{\circ}\)

Question 18

The radii of two circles with center at \(A\) and \(B\) are \(11\) cm and \(6\) cm respectively. If \(PQ\) is the common tangent of the circles and \(AB = 13\) cm, length of PQ is

A13 cm B12 cm C17 cm D8.5 cm

Question 19

Suppose \(Q\) is a point on the circle with centre \(P\) and radius \(1\), as shown in the figure; \(R\) is a point outside thr circle such that \(QR = 1\) and \(\angle QRP = 2^{\circ}\). Let \(S\) be the point where the segment \(RP\) intersects the given circle. Then measure of \(\angle RQS\) equals.

A\(86^{\circ}\) B\(87^{\circ}\) C\(88^{\circ}\) D\(89^{\circ}\)

Question 20

The chords \(PQ\) and \(RS\) of a circle are extended to meet at the point \(Q\). If \(PQ = 6\) cm, \(OQ = 8\) cm, \(OS = 7\) cm, then length of \(RS\)

A\(12\) cm B\(9\) cm C\(10\) cm D\(16\) cm

Question 21

\(ABC\) is a triangle in which \(AB = 4\) cm, \(BC = 5\) cm and \(AC = 6\) cm. A circle is drawn to touch side \(BC\) at \(P\), side \(AB\) extended at \(Q\) and side \(AC\) extended at \(R\). Then, \(AQ\) equals:

A\(7\) cm B\(7.5\) cm C\(6.5\) cm D\(15\) cm

Question 22

A line from one vertex \(A\) of an equilateral \(\triangle ABC\) meets the opposite side \(BC\) in \(P\) and the circumcircle of \(\triangle ABC\) in \(Q\).If \(BQ = 4\) cm and \(CQ = 3\) cm, then \(PQ\) is equal to

A\(7 \) cm B\(\frac{4}{3}\) cm C\(\frac{12}{7}\) cm D\(2 \) cm

Question 23

In the figure, \(AB, AC\) and \(BC\) are three tangents touching the circle at \(D\), \(E\) and \(F\) respectively. If \(AC = 24\) cm, \(BC = 18\) cm and \( \angle ACB = 90^{\circ}\),the radius of the circle is

A\(3\) cm B\(4\) cm C\(5\) cm D\(6\) cm

Question 24

In the figure, \(ST\) is a tangents to the smaller circle, \(ABC\) is a straight line. If \(\angle TAD = 2x\) and \(\angle DPC = 3x\), find \(x\)

A\(30^{\circ}\) B\(36^{\circ}\) C\(40^{\circ}\) D\(42^{\circ}\)

Question 25

In the adjoining figure, \(ABC\) is a triangle in which, \(\angle B = 90^{\circ}\) and its incircle \(C_1\) has radius \(3\) units. A circle \(C_2\) of radius \(1\) unit touches sides \(AC\), \(BC\) and the circle \(C_1\). Then length \(AB\) is equal to

A\(3+6\sqrt{3}\) B\(10+3\sqrt{2}\) C\(10+2\sqrt{3}\) D\(9+3\sqrt{3}\)

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