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Question
In the given figure, PQ is a tangent to the circle at A. AB and AD are bisectors of ∠CAQ and ∠PAC. If ∠BAQ = 30°, prove that : BD is diameter of the circle.
Solution
∠CAB = ∠BAQ = 30° ...(AB is angle bisector of ∠CAQ)
∠CAQ = 2∠BAQ = 60° ...(AB is angle bisector of ∠CAQ)
∠CAQ + ∠PAC = 180° ...(angles in linear pair)
∴ ∠PAC = 120°
∠PAC = 2∠CAD ...(AD is angle bisector of ∠PAC)
∠CAD = 60°
Now,
∠CAD + ∠CAB = 60° + 30° = 90°
∠DAB = 90°
Thus, BD subtends 90° on the circle
So, BD is the diameter of circle
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