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: 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

Proof: 

In ∆PAD and ∆QAD,

seg PA ≅ seg QA      .....[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  .....[Corresponding sides of congruent triangles]