Problem Statement
Prove that the lengths of tangents drawn from an external point to a circle are equal.
Verified Solution & Marking Scheme
State Construction
Let $P$ be external point, tangents touch at $Q$ and $R$. Join $OP, OQ, OR$. Tangents are perpendicular to radii: $\angle OQP = \angle ORP = 90^\circ$.
Apply RHS Congruence
In $\triangle OQP$ and $\triangle ORP$: hypotenuse $OP = OP$ (common), $OQ = OR$ (radii), $\angle OQP = \angle ORP = 90^\circ$. Hence $\triangle OQP \cong \triangle ORP$ by RHS.
Conclude CPCT
Therefore $PQ = PR$ by corresponding parts of congruent triangles.