Two tangents TP and TQ are drawn from an external point T to a circle with center O as shown in Fig. 10.73. If they are inclined to each other at an angle of 100°, then what is the value of ∠ POQ?

Question

Philoid · Accepted 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°