Question

In the following figure, 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, find ∠QSR.

Answer 1

Given: PQL and PRM are tangents at Q and R respectively, S is on the circle, ∠SQL = 50° and ∠SRM = 60°.

Step-wise calculation:

1. Use the tangent–chord theorem: the angle between a tangent and a chord through the point of contact equals the angle in the opposite (alternate) segment of the circle.

2. ∠SQL = 50° is the angle between tangent QL and chord QS, so the angle subtended by chord QS at the opposite point R on the circle is ∠QRS = 50°.

3. ∠SRM = 60° is the angle between tangent RM and chord RS, so the angle subtended by chord RS at the opposite point Q on the circle is ∠SQR = 60°.

4. In triangle QSR, angles are ∠QRS = 50° and ∠SQR = 60°, so ∠QSR = 180° – (50° + 60°) = 70°.