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

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#### Solution

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°.

Concept: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles

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