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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 = ?
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°