Question

In the given figure, O is the centre of the circle. Determine ∠AQB and ∠AMB, if PA and PB are tangents.

Answer 1

PA and PB are tangents to the circle.

∠APB = 75°.

Step 1: Find central angle AOB

Angle between two tangents = 180° − angle at the centre.

∴ ∠AOB = 180° − 75°

∠AOB = 105°

i. ∠AQB

∠AQB is an inscribed angle subtending the minor arc AB.

∠AQB = `1/2` × ∠AOB

∠AQB = `1/2`× 105°

∠AQB = 52.5°

ii. ∠AMB

Point M lies on the circle on the side of the major arc AB,
So ∠AMB subtends the major arc AB.

Major arc AB = 360° − 105° = 255°

∴ ∠AMB = `1/2` × 255°

∠AMB = 127.5°