If PA and PB are two tangents drawn to a circle with centre O from P such that PBA 50

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

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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From an external point P, tangents PA and PB are drawn to a circle with centre O. If ∠PAB = 50°, then find ∠AOB.

It is given that PA and PB are tangents to the given circle.

∴∠PAO=90°      (Radius is perpendicular to the tangent at the point of contact.)

Now

∠PAB=50°          (Given)

∴∠OAB=∠PAO−∠PAB=90°−50°=40°

In ∆OAB,

OB = OA    (Radii of the circle)

∴∠OAB=∠OBA=40°           (Angles opposite to equal sides are equal.)

Now

∠AOB+∠OAB+∠OBA=180°       (Angle sum property)

⇒∠AOB=180°−40°−40°=100°

Concept: Number of Tangents from a Point on a Circle

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