In Fig. 10.85, if PR is tangent to the circle at P and Q is the centre of the circle, then ∠POQ =

Question

In Fig. 10.85, if PR is tangent to the circle at P and Q is the centre of the circle, then ∠ POQ =

Philoid · Accepted Answer

Given:

∠RPQ = 60°

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 triangle = 180°.

By property 1, ∆OPR is right-angled at ∠OPR (i.e., ∠OPR = 90°).

OP = OQ [∵ radius of circle]

∴ ∠OPQ = ∠OQP = 30°

Now by property 2,

∠OPQ + ∠OQP + ∠POQ = 180°

⇒ 30° + 30° + ∠POQ = 180°

⇒ 60° + ∠POQ = 180°

⇒ ∠POQ = 180° - 60°

⇒ ∠POQ = 120°

Hence, ⇒ ∠POQ = 120°