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.

Answer 1

∠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