Question

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Answer 1

Let QR be a chord in a circle with center O and ∠1 and ∠2 are the angles made by tangent at point R and Q with chord respectively.

To Prove: ∠1 = ∠2

Let P be another point on the circle, then, join PQ and PR.

Since, at point Q, there is a tangent.

∠RPQ = ∠2   ...[Angles in alternate segments are equal] [Equation 1]

Since, at point R, there is a tangent.

∠RPQ = ∠1  ...[Angles in alternate segments are equal] [Equation 2]

From equation 1 and equation 2

∠1 = ∠2

Hence Proved.