In the given figure, PQ is a chord of a circle with center 0 and PT is a tangent. If ∠QPT = 60°, find ∠P

Given : , PQ is a chord of a circle with center 0 and PT is a tangent and ∠QPT = 60°.

To Find : ∠PRQ

∠OPT = 90°

∠OPQ + ∠QPT = 90°

∠OPQ + 60° = 90°

∠OPQ = 30° … [1]

Also.

OP = OQ [radii of same circle]

∠OQP = ∠OPQ [Angles opposite to equal sides are equal]

From [1], ∠OQP = ∠OQP = 30°

In △OPQ , By angle sum property

∠OQP + ∠OPQ + ∠POQ = 180°

30° + 30° + ∠POQ = 180°

∠POQ = 120°

As we know, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

So, we have

2∠PRQ = reflex ∠POQ

2∠PRQ = 360° - ∠POQ

2∠PRQ = 360° - 120° = 240°

∠PRQ = 120°

13