Answer :

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

To find : ∠PRQ

Now ∠OPT = 90° as OP⏊PT

[As the tangent drawn at any point on a circle is perpendicular to the radius through the point of contact]

Given ∠QPT = 60°

∠OPT - ∠OPQ = 60°

90° - ∠OPQ = 60°

∠OPQ = 30°

Now as OP = OQ [ radii of same circle]

∠OQP = ∠QPQ = 30° [angle opposite to equal sides are equal]

In triangle OPQ

∠OPQ + ∠POQ + ∠OQP = 180° [Angle sum property of a triangle]

30 + ∠POQ + 30 = 180

∠POQ = 120°

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,

Reflex ∠POQ = 360 - 120 = 240°

Now PQ is an arc and we know

Reflex ∠POQ = 2∠PRQ

∠PRQ = 240/2 = 120°

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