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Solutions

Overview

Step-by-step NCERT solutions for Circles (Chapter 10, CBSE Class 10 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Circles textbook chapter.

Solved

What these solutions cover

All 17 questions in Circles are solved in the PDF. Here's what's inside, exercise by exercise:

Tangent to a Circle — Introduction

  1. How many tangents can a circle have?
  2. Fill in the blanks:
    • (i) A tangent to a circle intersects it in ___ point(s).
    • (ii) A line intersecting a circle in two points is called a ___.
    • (iii) A circle can have ___ parallel tangents at the most.
    • (iv) The common point of a tangent to a circle and the circle is called ___.
  3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is: (A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm
  4. Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Properties of Tangents

  1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is: (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm
  2. In the given figure, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to: (A) 60° (B) 70° (C) 80° (D) 90°
  3. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠POA is equal to: (A) 50° (B) 60° (C) 70° (D) 80°
  4. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
  5. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
  6. The length of a tangent from a point A at distance 5√2 cm from the centre of the circle is 5 cm. Find the radius of the circle.
  7. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
  8. A quadrilateral ABCD is drawn to circumscribe a circle (see figure). Prove that AB + CD = AD + BC.
  9. In the given figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. Prove that ∠AOB = 90°.
  10. 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-segment joining the points of contact at the centre.
  11. Prove that the parallelogram circumscribing a circle is a rhombus.
  12. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC.
  13. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Sample

Worked examples

2 questions from Circles, solved step by step.

Tangent to a Circle — Introduction

How many tangents can a circle have?

A tangent to a circle is a line that touches the circle at exactly one point.

At every point on the circle, there is exactly one tangent.

Since a circle has infinitely many points on its circumference, a circle can have infinitely many tangents.

Answer: A circle can have infinitely many tangents.

Properties of Tangents

The length of a tangent from a point A at distance 5√2 cm from the centre of the circle is 5 cm. Find the radius of the circle.

Let O be the centre and AT be the tangent touching at T.

Given: OA = 5√2 cm, AT = 5 cm.

Since AT is tangent at T, OT ⊥ AT.

∴ Triangle OTA is right-angled at T.

By Pythagoras theorem:

OA² = OT² + AT²

(5√2)² = OT² + 5²

25 × 2 = OT² + 25

50 = OT² + 25

OT² = 50 − 25 = 25

OT = 5 cm

∴ The radius of the circle is 5 cm.

Answer: Radius = 5 cm

All 17 questions in Circles are solved step by step in the Solutions PDF above.

Board papers

Circles in CBSE board papers

Tested in 113 questions across 27 Class 10 board papers (2020–2026). Open a set to see exactly how this chapter is asked:

2026
Set 44 QsSet 53 Qs
2025
Set 15 QsSet 24 QsSet 34 QsSet 44 QsSet 55 QsSet 64 Qs
2024
Set 13 QsSet 25 QsSet 34 QsSet 44 QsSet 54 Qs
2023
Set 17 QsSet 26 QsSet 44 QsSet 55 QsSet 63 Qs
2022
Set 13 QsSet 24 QsSet 35 QsSet 42 Qs
2020
Set 13 QsSet 22 QsSet 36 QsSet 45 QsSet 55 Qs
Keep solving

More solutions in Mathematics

01Real Numbers02Polynomials03Pair of Linear Equations in Two Variables04Quadratic Equations05Arithmetic Progressions06Triangles07Coordinate Geometry08Introduction to Trigonometry09Some Applications of Trigonometry11Areas Related to Circles12Surface Areas and Volumes13Statistics14Probability
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More NCERT Solutions for Class 10

Read the Circles textbook chapter / PDF, or browse all CBSE Class 10 Mathematics solutions.

← PreviousCh 9: Some Applications of TrigonometryNext →Ch 11: Areas Related to Circles

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