Question

In the given figure, PA and PB are two tangents to the circle with centre O. If ∠APB = 60° then find the measure of ∠OAB.

Answer 1

Construction: Join OB

We know that the radius and tangent are perpendicular at their point of contact

∵ ∠OBP = ∠OAP = 90°
Now, In quadrilateral AOBP
∠AOB + ∠OBP + ∠APB +∠OAP = 360°        [Angle sum property of a quadrilateral]
 ⇒ ∠AOB +90° + 60° + 90° = 360°
⇒ 240° + ∠AOB = 360°
⇒ ∠ AOB = 120°
Now, In isosceles triangle AOB
∠AOB  + ∠OAB + ∠OBA = 180°                [Angle sum property of a triangle]
⇒ 120° +  2 ∠OAB =180°               [∵ ∠OAB = ∠OBA]
⇒  ∠OAB = 30°