Question

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

Answer 1

The attached figure shows two tangents, $SK$ and $SR$ drawn to circle with center $O$ from an external point $K$.

To prove that: $SK=RK$

Proof:

Normal and tangent at a point on the circle are perpendicular to each other.

$\angle OSK=\angle ORK={90}^o$

Using Pythagoras Theorem,

$O{K}^2=O{S}^2+S{K}^2............\left(i\right)$

$O{K}^2=O{R}^2+R{K}^2............\left(ii\right)$

Subtracting (ii) from (i),

$O{K}^2-O{K}^2=O{S}^2+S{K}^2-O{R}^2-R{K}^2$

$⟹S{K}^2=R{K}^2\because OS=OR$

$SK=RK$

Hence, proved