In Fig. 9.6, if O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to
a. 100°
b. 80°
c. 90°
d. 70°

Solution:

Given, O is the centre of a circle.

PQ is a chord

PR is a tangent at P which makes an angle of 50° with PQ.

We have to find the measure of angle POQ.

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

i.e., ∠OPR = 90°

Given, ∠RPQ = 50°

We know that ∠OPR = ∠OPQ + ∠RPQ

90° = ∠OPQ + 50°

∠OPQ = 90° - 50°

∠OPQ = 40°

We know that if the opposite sides are equal then the opposite angles are equal.

From the figure,

OP = OQ = Radius

So, ∠OPQ = ∠OQP = 40°

In triangle POQ,

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

∠OPQ + ∠OQP + ∠POQ = 180°

40° + 40° + ∠POQ = 180°

80° + ∠POQ = 180°

∠POQ = 180° - 80°

Therefore, ∠POQ = 100°

✦ 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

Given, PT is tangent to the circle with centre O.

PQ is a chord

∠QPT = 70°

We have to find the measure of ∠POQ.

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

So, ∠OPT = 90°

From the figure,

∠OPQ + ∠QPT = 90°

∠OPQ + 70° = 90°

∠OPQ = 90° - 70°

∠OPQ = 20°

We know, OP = OQ = radius

In triangle OPQ,

OP and OQ are equal.

So, OPQ is an isosceles triangle.

Now, ∠OPQ = ∠OQP = 20°

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

∠POQ + ∠OPQ + ∠OQP = 180°

∠POQ + 2∠OPQ = 180°

∠POQ + 2(20°) = 180°

∠POQ = 180° - 40°

Therefore, ∠POQ = 140°

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

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NCERT Exemplar Class 10 Maths Exercise 9.1 Problem 7

In Fig. 9.6, if O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to a. 100°, b. 80°, c. 90°, d. 70°

Summary:

In Fig. 9.6, if O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to 100°

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☛ Related Questions:

• In Fig. 9.7, if PA and PB are tangents to the circle with centre O such that ∠APB = 50°, then ∠OAB i . . . .
• If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then length of each t . . . .
• In Fig. 9.8, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR . . . .