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Question
Prove the following theorem:
Tangent segments drawn from an external point to the circle are congruent.
Solution
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 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]
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