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In figure, PQ and PR are tangents to the circle with center O and S is a point on the circle such that ∠SQL = 50 degree and ∠SRM = 60 degree. Find ∠QSR.

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70^{\circ}

60^{\circ}

50^{\circ}

40^{\circ}

answer is A.

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Detailed Solution

First of all, join QR and redraw the image,
Question Image

In this diagram we have,

∠ QSR = ∠ SQL = 50 degrees

(as angle equal to opposite sides of isosceles triangles are equal)
Also,

∠ RQS = ∠ SRM = 60 degrees

Now, it is known that angle formed by a chord and a tangent is equal to the angle that chord subtends at the circumference of segment that is opposite the angle.
Therefore,

∠ QSR = 180 - (60 + 50) = 70 Degrees

(by angle sum property of triangle)
Also,

∠ OSQ = ∠ OQS = 90^{\circ} - 50^{\circ} = 40^{\circ}

Similarly,

∠ RSQ = ∠ SRO = 90^{\circ} - 60^{\circ} = 30^{\circ}

Now, from the figure,

∠ QSR = ∠ OSQ + ∠ RSQ

∠ QSR = 40^{\circ} + 30^{\circ}

∠ QSR = 70^{\circ}

Correct option is 1.

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