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In Fig. 9.2, PQ is a chord of a circle and PT is the tangent at P such that ∠QPT = 60°. Then ∠PRQ is equal to
a. 135°
b. 120°
c. 115°
d. 150°

In Fig. 9.2, PQ is a chord of a circle and PT is the tangent at P such that ∠QPT = 60°. Then ∠PRQ is equal to

Solution:

Given, PQ is a chord of a circle

PT is the tangent at P

Also, ∠QPT = 60°

We have to find the angle PRQ.

We know that the radius is perpendicular to the tangent at the point of contact.

OP is the radius

OP ⟂ PT

So, ∠OPQ + ∠QPT = 90°

∠OPQ + 60° = 90°

∠OPQ = 90° - 60°

∠OPQ = 30°

OP = OQ = radius

So ∠OQP = 30°

In triangle POQ,

We know that the sum of all three interior angles of a triangle is equal to 180°

∠POQ + ∠OPQ + ∠OQP = 180°

∠POQ + 30° + 30° = 180°

∠POQ + 60° = 180°

∠POQ = 180° - 60°

∠POQ = 120°

Now, ∠POQ + reflex angle of ∠POQ = 360°

120° + reflex angle of ∠POQ = 360°

reflex angle of ∠POQ = 360° - 120°

reflex angle of ∠POQ = 240°

We know that the angle subtended by an arc at the centre doubles the angle subtended by the arc at any point on the remaining part of the circle.

∠PRQ = 1/2(∠POQ)

∠PRQ = 1/2(240°)

∠PRQ = 120°

Therefore, the angel PRQ is equal to 120°

✦ Try This: If PT is a tangent to a circle with centre O and PQ is a chord of the circle such that ∠QPT = 70°, then find the measure of ∠POQ.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10

NCERT Exemplar Class 10 Maths Exercise 9.1 Sample Problem 3

In Fig. 9.2, PQ is a chord of a circle and PT is the tangent at P such that ∠QPT = 60°. Then ∠PRQ is equal to a. 135°, b. 120°, c. 115°, d. 150°

Summary:

In Fig. 9.2, PQ is a chord of a circle and PT is the tangent at P such that ∠QPT = 60°. Then ∠PRQ is equal to 120°

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