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Class 10 Circles: 8 practice questions with answers, step by step

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These are 8 practice questions on NCERT Class 10 Maths Chapter 10, Circles, each with a worked answer and the textbook page it comes from. They cover how many tangents a point allows, the radius meeting the tangent at a right angle, why the tangent is shorter than the line to the centre, a mistake about a radius, the distance from the centre, equal tangents from an outside point, the angles in the tangent figure, and a perimeter made of tangents. Try each one on paper before you open the answer.

These questions are our own, drafted with AI help. Every answer was re-computed by a test script, but no practising teacher has reviewed them yet. If you find a mistake, please tell us.

What the chapter says you need

Facts from the NCERT textbook (Mathematics, Textbook for Class X, Reprint 2026-27, Chapter 10, printed pages 144 to 153):

Points, tangents and the radius

Question 1. A circle has centre O and radius 58 mm. Point A is 41 mm from O, point B is 58 mm from O and point C is 73 mm from O. How many tangents to the circle can be drawn through each point?

Answer: none through A, one through B, two through C. A is closer to O than the radius, so it lies inside the circle and no tangent passes through it. B is exactly one radius from O, so it lies on the circle and there is one and only one tangent there. C is farther than the radius, so it lies outside the circle and exactly two tangents can be drawn from it. (Number of tangents from a point, page 148.)

Question 2. The tangent from a point Q to a circle is 55 cm long, and Q is 73 cm from the centre. Ravi says the radius must be 73 cm minus 55 cm. What is wrong with his reasoning, and what is the radius?

Answer: the lengths do not subtract; the radius is 48 cm. The radius to the point of contact is perpendicular to the tangent (Theorem 10.1), so the radius, the tangent and the line from Q to the centre form a right triangle, with the 73 cm line as its hypotenuse. In a right triangle it is the squares of the sides that add, not the sides themselves: radius² = 73² − 55² = 5329 − 3025 = 2304, so the radius is 48 cm (48 × 48 = 2304). (Theorem 10.1, pages 146 and 147.)

Question 3. A circular park has radius 51 m. A straight road touches the park's boundary at point P. A bus stop on the road is 68 m from P. How far is the bus stop from the centre of the park?

Answer: 85 m. The road is a tangent at P, so the radius to P is perpendicular to the road (Theorem 10.1). The distance from the centre to the bus stop is the hypotenuse of a right triangle with legs 51 m and 68 m: √(51² + 68²) = √(2601 + 4624) = √7225 = 85 m (85 × 85 = 7225). (Theorem 10.1, pages 146 and 147.)

Equal tangents

Question 4. Assertion (A): From a point outside a circle, the two tangents drawn to it are equal in length. Reason (R): The two right triangles formed by the centre, the outside point and the two points of contact are congruent by RHS. Choose the correct option: both true and R explains A; both true but R does not explain A; A true and R false; or A false and R true.

Answer: A and R are both true, and R explains A. The radii are equal, the line from the centre to the outside point is common, and both angles at the points of contact are 90° (Theorem 10.1), so the two triangles are congruent by RHS and the tangents are equal. (Theorem 10.2, page 149.)

Question 5. Explain why the tangent from a point outside a circle is always shorter than the distance from that point to the centre. Check your reason with a circle of radius 28 cm and a point T that is 53 cm from the centre.

Answer: the line to the centre is the hypotenuse of a right triangle, so it is the longest side. If the tangent from T touches the circle at P, then OP ⊥ TP (Theorem 10.1), so triangle OPT is right-angled at P and OT is its hypotenuse. The hypotenuse is the longest side of a right triangle, so the tangent TP is shorter than OT. Check: TP² = 53² − 28² = 2809 − 784 = 2025, so TP = 45 cm, which is less than 53 cm. (Theorem 10.1, pages 146 and 147; length of a tangent, page 148.)

Two tangents, angles and perimeters

Question 6. Two tangents TP and TQ touch a circle with centre O and radius 33 cm, and TP = 56 cm. What is TQ? Why is the four-sided figure OPTQ a kite, and what is its perimeter?

Answer: TQ = 56 cm; OPTQ is a kite; perimeter 178 cm. By Theorem 10.2, the two tangents from T are equal, so TQ = 56 cm. In OPTQ, OP = OQ = 33 cm (radii) and TP = TQ = 56 cm, so it has two pairs of equal adjacent sides: a kite, with right angles at P and Q (Theorem 10.1). Perimeter = 33 + 56 + 56 + 33 = 178 cm. (Theorems 10.1 and 10.2, pages 146 to 149.)

Question 7. Two tangents TP and TQ are drawn from an external point T to a circle with centre O, and ∠PTQ = 46°. Find ∠TPQ and ∠OPQ.

Answer: ∠TPQ = 67°; ∠OPQ = 23°. By Theorem 10.2, TP = TQ, so triangle TPQ is isosceles and its base angles are equal: ∠TPQ = ∠TQP = (180° − 46°) ÷ 2 = 134° ÷ 2 = 67°. By Theorem 10.1, ∠OPT = 90°, so ∠OPQ = 90° − 67° = 23°, which is half of ∠PTQ, as the chapter's Example 2 shows in general. (Theorems 10.1 and 10.2, pages 146 to 149; Example 2, page 150.)

Question 8. From a point P outside a circle, two tangents touch the circle at A and B, and PA = 47 cm. A third tangent touches the circle at a point C on the arc between A and B that faces P, and meets PA at X and PB at Y. Find the perimeter of triangle PXY.

Answer: 94 cm. By Theorem 10.2, the two tangents from each outside point are equal: PB = PA = 47 cm, XC = XA and YC = YB. So XY = XC + CY = XA + YB, and the perimeter is PX + XY + YP = (PX + XA) + (YB + YP) = PA + PB = 47 + 47 = 94 cm, wherever C lies on that arc. (Theorem 10.2, page 149.)

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Sources

All opened and read on 21 October 2026. Page numbers are the printed page numbers of the textbook.

The definitions, theorems and examples are from the textbook pages above. The questions, worked answers and checks are our own, and every number was re-computed by script. This guide is general information; follow your teacher and the official CBSE notices.