Question

In the given figure, PA and PB are two tangents to the circle with centre O. If ∠APB = 50^∘ then what is the measure of ∠OAB is [CBSE 2015]

Answer 1

Construction: Join OB


We know that the radius and tangent are perperpendular 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^∘ + 50^∘+ 90^∘ = 360^∘
⇒ 230^∘+ ∠BOC = 360^∘
⇒ ∠AOB = 130^∘
Now, In isoceles triangle AOB
∠AOB + ∠OAB + ∠OBA = 180^∘ [Angle sum property of a triangle]
⇒ 130^∘ + 2∠OAB = 180⁰ [∵∠OAB = ∠OBA ]
⇒ ∠OAB = 25^∘