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Two tangents TP and TQ are drawn from an external point T to a circle with center O as shown in Fig. If they are inclined to each other at an angle of 100°, then what is the value of ∠ POQ?

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Best answer

Given:

∠QTP = 100°

Property 1: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency.

Property 2: Sum of all angles of a quadrilateral = 360°.

By property 1,

∠OPT = 90° and ∠OQT = 90°

And,

By property 2,

∠QTP + ∠OPT + ∠OQT + ∠POQ = 360° 

⇒ ∠POQ = 360° – ∠QTP + ∠OPT + ∠OQT 

⇒ ∠POQ = 360° – 100° + 90° + 90° 

⇒ ∠POQ = 360° – 280° 

⇒ ∠POQ = 80°

Hence, 

∠POQ = 80°

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