Question

In the given figure, PQ is a tangent to a circle with centre O. A is the point of contact. If ∠PAB = 67°, then the measure of ∠AQB is ______.

Options

• 73°

• 64°

• 53°

• 44°

Answer 1

In the given figure, PQ is a tangent to a circle with centre O. A is the point of contact. If ∠PAB = 67°, then the measure of ∠AQB is 44°.

Explanation:

∠BAC = 90°   ...[Angle in a semicircle]

∠PAB + ∠BAC + ∠CAQ = 180°

⇒ ∠CAQ = 180° – (90° + 67°)

⇒ ∠CAQ = 23°

∠ACB = ∠PАB = 67°   ...[Angles in alternate segments]

∠ACB + ∠ACQ = 180°   ...[Linear pair]

⇒ ∠ACQ = 180° – 67° 

⇒ ∠ACQ = 113°

Now, ∠CAQ + ∠ACQ + ∠AQC = 180°   ...[In ΔACQ]

⇒ ∠AQC = 180° – (23° + 113°) = 44° 

⇒ ∠AQB = 44°