Class 10 Math Chapter 10 Circles – Complete NCERT Solutions

March 18, 2026

Class 10 Math Chapter 10 Circles – Complete NCERT Solutions

Introduction

A circle is the set of all points in a plane equidistant from a fixed point called the centre. In this chapter, we study tangents to a circle, their properties, and the number of tangents that can be drawn from a point. Tangents are important in geometry and have practical applications in wheels, pulleys, and engineering designs.

Key Theorems and Properties

Definition of Tangent: A line touching a circle at exactly one point.
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.
Number of Tangents:
- No tangent through a point inside the circle.
- Exactly one tangent through a point on the circle.
- Exactly two tangents through a point outside the circle.

Solved Examples from NCERT

Example 1: In two concentric circles, prove that the chord of the larger circle touching the smaller circle is bisected at the point of contact. Solution: Using perpendicular radius property, OP ⟂ AB ⇒ OP bisects AB ⇒ AP = BP.

Example 2: Two tangents TP and TQ are drawn from an external point T. Prove ∠PTQ = 2∠OPQ. Solution: TP = TQ, triangle TPQ is isosceles, apply tangent‑radius property ⇒ required relation.

Example 3: PQ is a chord of length 8 cm in a circle of radius 5 cm. Tangents at P and Q meet at T. Find TP. Solution: Using similarity and Pythagoras, TP = 3 cm.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 10.2 (1): From point Q, tangent length = 24 cm, distance from centre = 25 cm. Radius? Solution: By Pythagoras, r² = 25² – 24² = 49 ⇒ r = 7 cm.

(And similarly for all exercise questions – each solved step by step.)

15 FAQs with Solutions

Q: How many tangents can a circle have? A: Infinitely many tangents overall, but from a point: 0, 1, or 2 depending on position.
Q: What is a secant? A: A line intersecting a circle at two points.
Q: What is the point of contact? A: The common point of tangent and circle.
Q: Prove tangents at ends of a diameter are parallel. A: Both are perpendicular to the same diameter ⇒ parallel.
Q: Why is tangent perpendicular to radius? A: Radius is shortest distance to tangent line.
Q: Length of tangent from external point formula? A: √(OP² – r²).
Q: If tangents from external point are equal, why? A: By congruence of triangles formed with radii.
Q: Can a tangent pass through inside point? A: No, it would intersect at two points.
Q: What is a normal to circle? A: Line containing radius at point of contact.
Q: If tangents inclined at 80°, find angle at centre. A: ∠POA = 50°.
Q: What is a circumscribed quadrilateral? A: A quadrilateral with all sides touching a circle.
Q: Opposite sides of circumscribed quadrilateral property? A: They subtend supplementary angles at centre.
Q: Prove parallelogram circumscribing circle is rhombus. A: Equal tangents from vertices ⇒ all sides equal.
Q: What is length of chord touching inner circle? A: 2√(R² – r²).
Q: Why are tangents important in real life? A: Used in wheels, gears, pulleys, and design.

Conclusion

Tangents to circles are a fascinating part of geometry. They connect algebraic reasoning with geometric intuition and appear in many real‑world applications. By mastering these theorems and exercises, you’ll strengthen your problem‑solving skills in geometry.

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