Question

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Answer 1

Given:

TP and TQ are two tangents drawn from an external point T to the circle C (O, r).

To prove: TP = TQ

Construction: Join OT.

Proof:

We know that a tangent to the 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)

⇒ TP = TQ  ...(CPCT)

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