Question

Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

Answer 1

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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]

See the diagram in picture

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