How to prove that the length of tangents drawn from an external point to a circle are equal.

Dear Student!

Given: PT and TQ are two tangent drawn from an external point T to the circle C (O, r).

To prove: PT = TQ

Construction: Join OT.

Proof: We know that, a tangent to circle is perpendicular to the radius through the point of contact.

∴ ∠OPT = ∠OQT = 90°

In ΔOPT and ΔOQT,

OT = OT  (Common)

OP = OQ  ( Radius of the circle)

∠OPT = ∠OQT  (90°)

∴ ΔOPT ΔOQT  (RHS congruence criterion)

⇒ PT = TQ  (CPCT)

Thus, the lengths of the tangents drawn from an external point to a circle are equal.

Cheers!