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In Fig. 9.10, PQL and PRM are tangents to the circle with centre O at the points Q and R, respectively and S is a point on the circle such that ∠SQL = 50° and ∠SRM = 60°. Then ∠QSR is equal to 40°. Write ‘True’ or ‘False’ and give reasons for your answer

Solution:

Given, PQL and PRM are the tangents to the circle with centre O at the points Q and R

S is a point on the circle such that ∠SQL = 50° and ∠SRM = 60°

We have to determine if ∠QSR is equal to 40°.

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

So, ∠LQO = ∠MRO = 90°

From the figure,

∠LQO = ∠LQS + ∠SQO

90° = 50° + ∠SQO

∠SQO = 90° - 50°

∠SQO = 40°

From the figure,

∠MRO = ∠MRS + ∠SRO

90° = 60° + ∠SRO

∠SRO = 90° - 60°

∠SRO = 30°

Considering triangle OSQ,

OS = OQ = radius

Two sides of a triangle are equal

So, OSQ is an isosceles triangle.

In an isosceles triangle, two sides and two angles are equal.

Two equal angles are ∠SOQ = ∠ORS

∠QSR = ∠OSQ + ∠OSR

∠QSR = 30° + 40°

Therefore, ∠QSR = 70°

✦ Try This: PA and PB are two tangents to the circle with centre O. If ∠APB = 60° then find the measure of ∠OAB.

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

NCERT Exemplar Class 10 Maths Exercise 9.2 Sample Problem 2

In Fig. 9.10, PQL and PRM are tangents to the circle with centre O at the points Q and R, respectively and S is a point on the circle such that ∠SQL = 50° and ∠SRM = 60°. Then ∠QSR is equal to 40°. Write ‘True’ or ‘False’ and give reasons for your answer

Summary:

The statement “In Fig. 9.10, PQL and PRM are tangents to the circle with centre O at the points Q and R, respectively and S is a point on the circle such that ∠SQL = 50° and ∠SRM = 60°. Then ∠QSR is equal to 40°” is false

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