Mathematics
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.
Circles
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Answer
From figure,
∠CAB = ∠BAQ = 30° (AB is angle bisector of ∠CAQ)
⇒ ∠CAQ = 2∠BAQ = 60°.
From figure,
⇒ ∠CAQ + ∠PAC = 180° [Linear pair]
⇒ 60° + ∠PAC = 180°
⇒ ∠PAC = 180° - 60°
⇒ ∠PAC = 120°.
⇒ ∠PAC = 2∠CAD (AD is angle bisector of ∠PAC)
⇒ 120° = 2∠CAD
⇒ ∠CAD =
⇒ ∠CAD = 60°.
From figure,
∠DAB = ∠CAD + ∠CAB = 60° + 30° = 90°.
Thus BD, subtends 90° on the circle. Since, angle in semi-circle is a right angle.
Hence, BD is the diameter of the circle.
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