Q.3 (B)(2) Prove that tangent segments drawn from an external point to a circle are congruent.

Answer: Proof by the hypotenuse-side test (Δ PAO ≅ Δ PBO).

Given: a circle with centre O; P is an external point; PA and PB are tangent segments touching the circle at A and B.

To prove: seg PA ≅ seg PB.

Construction: draw radii OA and OB and segment OP.

Proof: ∠OAP = ∠OBP = 90º (a tangent is perpendicular to the radius at the point of contact).

In right triangles PAO and PBO: hypotenuse OP ≅ OP (common side), and OA ≅ OB (radii of the same circle).

So Δ PAO ≅ Δ PBO (hypotenuse-side test), and seg PA ≅ seg PB (corresponding sides of congruent triangles).