Two tangents TP and TQ are drawn to a circle with center O from an external point T. Prove that PTQ = 2OPQ.

Question

Philoid · Accepted Answer

Let TP and TQ are two tangents of a circle at points P and Q respectively with center O.

To prove: ∠PTQ = 2∠OPQ

Let ∠PTQ = θ

As lengths of tangents drawn from an external point to the circle are equal, therefore TP = TQ.

∴ ΔPQT is an isosceles triangle.

∴ ∠TPQ = ∠TQP = 1/2 (180° - θ) = 90° - (θ/2)

Also, tangent at any point of a circle is perpendicular to the radius through the point of contact.

∴ ∠ OPQ = ∠OPT - ∠TPQ

= 90° - (90°- (θ/2))

= 1/2 (θ)

= 1/2 ∠PTQ

Thus, ∠PTQ = 2∠OPQ.

Hence, proved.