Question

If two tangents are drawn to a circle from an external point, show that they subtend equal angles at the centre.

Answer 1

Given: A circle with centre O and a point A outside it. Also, AP and AQ are the two tangents to the circle

To prove: ∠AOP = ∠AOQ

Proof: In ΔAOP and ΔAOQ, we have

AP = AQ   ...[Tangents from an external point are equal]

OP = OQ   ...[Radii of the same circle]

OA = OA   ...[Common side]

∴ ΔAOP ≅ ΔAOQ   ...[By SSS - congruence]

Hence, ∠AOP = ∠AOQ (c.p.c.t).