Question

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

Answer 1

Let a circle with centre $O$ and external point $P$.

Two tangents $PQ$ and $PR$ are drawn.

To prove : $PQ=PR$

Construction: Join radius $OQ$ and $OR$ also join $O$ to $P$.

Proof : In $△OQP$ and $△ORP$

$OQ=OR$ $\left(\text{Radii}\right)$

$\angle Q=\angle R$ $[\text{Each}{90}^{\circ },radius\perp tangents)$ ]

$OP=OP$ $\left(\text{Common side}\right)$

$\therefore △OQP$ $\cong △ORP$ $(RHS\text{Congruence rule}.)$

By CPCT $PQ=PR$

Hence, proved.