In the given figure, O is the center of a circle; PQL and PRM are the tangents at the points Q and R respectively and S is a point on the circle such that ∠SQL = 50° and ∠SRM = 60°. Then, ∠QSR = ?

Question

Philoid · Accepted Answer

As PL and PM are tangents to given circle,

We have,

OR ⏊ PM and OQ ⏊ PL

[Tangents drawn at a point on circle is perpendicular to the radius through point of contact]

So, ∠ORM = ∠OQL = 90°

∠ORM = ∠ORS + ∠SRM

90° = ∠ORS + 60°

∠ORS = 30°

And ∠OQL = ∠OQS + ∠SQL

90° = ∠OQS + 50°

∠OQS = 40°

Now, In △SOR

OS = OQ [radii of same circle]

∠ORS = ∠OSR

[Angles opposite to equal sides are equal]

∠OSR = 30°

[as ∠ORS = 30°]

Now, In △SOR

OS = SQ [radii of same circle]

∠OQS = ∠OSQ

[Angles opposite to equal sides are equal]

∠OSQ = 40° [as ∠OQS = 40°]

As,

∠QSR = ∠OSR + ∠OSQ

∠QSR = 30° + 40° = 70°