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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.
Solution
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°
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