Theorems on Tangent to a Circle | Proofs & Examples

A tangent to a circle is a straight line that touches the circle at exactly one point. It does not pass through the interior of the circle. It does not intersect the circle at two places. It arrives, makes contact at one single point, and moves on. Understanding the theorems related to tangents helps in solving many geometric problems efficiently. In this guide, you will learn about important theorems of tangents to a circle and their proofs.

Table of Contents

• Theorem 1: Tangent is perpendicular to the radius
• Theorem 2: Lengths of tangents from an external point

Theorem 1: Tangent is perpendicular to the radius

Statement: The tangent at any point of a circle is perpendicular to the

radius through the point of contact.

Proof: Given a circle with centre O and a tangent XY to the circle at a point P. To prove that OP is perpendicular to XY. Take a point Q on XY other than P; the point Q must lie outside the circle. Join OQ.

Therefore, OQ is longer than the radius OP of the circle, i.e., OQ > OP.

This happens for every point on the line XY except P; hence, OP is the shortest of all the distances from O to the points on XY. OP is perpendicular to XY (∵ the shortest distance from that point to the line is the perpendicular from the point to the line). ∴ The tangent at any point of a circle is perpendicular to the radius through the point of contact.

OQ² = OP² + QP²
13² = r² + 12²

169 = r² + 144
r² = 25
r = 5 cm

∴ radius = 5 cm.

Theorem 2: Lengths of tangents from an external point

Statement: The lengths of tangents drawn from an external point to a circle are equal.

Proof: Let O be the of the circle and P be a point lying outside the circle. Two tangents, PQ and PR from P, meet the circle at Q and R, respectively.

Draw: Join OP, OQ and OR.

∠ OQP and ∠ ORP are right angles, because these are angles between the radii and tangents.

Now in right triangles OQP and ORP,

OQ = OR (Radii of the same circle)

OP = OP (Common)

Therefore, ∆ OQP ≅ ∆ ORP (RHS congruency)

This gives PQ = PR (CPCT)

NOTE: ∆ OQP ≅ ∆ ORP ∴ ∠ OPQ = ∠ OPR. i.e., OP is the angle bisector of ∠QPR. The line passing through the r and the external point bisects the angle between the two tangents.

In △OAP: ∠OAP = 90° (Theorem 1), ∠APO = 30°

∠POA = 180° − (90° + 30°) = 60°