Question

prove that the tangent at the extremist of any chord make equal angles with the chord

Answer 1

Let PQ be the chord of a circle with center O

Let AP and AQ be the tangents at points P and Q respectively.

Let us assume that both the tangents meet at point A.

Join points O, P. Let OA meets PQ at R

Here we have to prove that ∠APR = ∠AQR

Consider, ΔAPR and ΔAQR

AP = AQ [Tangents drawn from an internal point to a circle are equal]

∠PAR = ∠QAR

AR = AR [Common side]

∴ ΔAPR ≅ ΔAQR [SAS congruence criterion]

Hence ∠APR = ∠AQR [CPCT]

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