Question

In Fig. 1, PA and PB are tangents to the circle with centre O such that ∠APB = 50°. Write the measure of ∠OAB.

Answer 1

PA and PB are tangents drawn from an external point P to the circle.

∴ PA = PB      (Length of tangents drawn from an external point to the circle are equal.)

In ∆PAB,

PA = PB

⇒ ∠PBA = ∠PAB     .....(1)     (Angles opposite to equal sides are equal.)

Now,

∠APB + ∠PBA + ∠PAB = 180°

⇒ 50º + ∠PAB + ∠PAB = 180°    [Using (1)]

⇒ 2∠PAB = 130°

⇒ ∠PAB =`130^@/2`= 65°

We know that radius is perpendicular to the tangent at the point of contact.

∴ ∠OAP = 90°         (OA ⊥ PA)

⇒ ∠PAB + ∠OAB = 90° 

⇒ 65° + ∠OAB = 90°

⇒∠OAB = 90° − 65° = 25°

Hence, the measure of ∠OAB is 25°.