Question

Theorem: Tangent segments drawn from an external point to a circle are congruent 

Draw radius AP and radius AQ and complete the following proof of the theorem. 

Answer 1

Given: A is the centre 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 and seg AQ.

Proof: 

In ∆PAD and ∆QAD, 

seg PA ≅ [segQA] [Radii of the same circle] 

seg AD ≅ seg AD [Common side]

∠APD = ∠AQD = 90° [Tangent theorem] 

∴ ∆PAD = ∆QAD [By Hypotenuse side test] 

∴ seg DP = seg DQ [c.s.c.t]