Question

Tangent segments drawn from an external point to a circle are congruent, prove this theorem. Complete the following activity.

Given: `square`

To Prove: `square`

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

In ∆PAD and ∆QAD,

seg PA ≅ `square`   ...[Radii of the same circle]

seg AD ≅ seg AD   ...[`square`]

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

∴ ∆PAD ≅ ∆QAD   ...[`square`]

∴ seg DP ≅ seg DQ   ...[`square`]

Answer 1

Given: \[\boxed{\text{A is the centre of the circle. Tangents through external point D touch the circle at the points P and Q.}}\]

To Prove: \[\boxed{\text{seg DP ≅ seg DQ}}\]

Proof: 

In ∆PAD and ∆QAD,

seg PA ≅ \[\boxed{\text{seg QA}}\]   ...[Radii of the same circle]

seg AD ≅ seg AD   ...\[\boxed{\text{[Common side]}}\]

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

∴ ∆PAD ≅ ∆QAD   ...\[\boxed{\text{[By Hypotenuse side test]}}\]

∴ seg DP ≅ seg DQ   ...\[\boxed{\text{[Corresponding sides of congruent triangles]}}\]