Question

Prove the following theorem:

Tangent segments drawn from an external point to the circle are congruent.

Prove that tangent segments drawn from an external point to a circle are congruent.

Answer 1

Given: A is the center of the circle. Tangents through external point D touch the circle at the points P and Q.

To prove: seg DP ≅ seg DQ

Construction: Draw seg AP, seg AD, and seg AQ.

Proof: In ΔPAD and ΔQAD,

seg PA ≅ seg QA    ...[Radii of the same circle]

seg AD ≅ seg AD    ...[Common side]

∠APD = ∠AQD = 90°    ...[Tangent theorem]

∴ ΔPAD ≅ ΔQAD    ...[By Hypotenuse side test]

∴ seg DP ≅ seg DQ  ...[Corresponding sides of congruent triangles]