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Q.

In the given figure, O is the centre 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=50⁰ and ∠SRM=60⁰. Then, which of the following is the measure of ∠QSR?


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a

40⁰

b

50⁰

c

60⁰

d

70⁰ 

answer is D.

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Detailed Solution

Given that PQL and PRM are the tangents at the points Q and R respectively and S is a point on the circle such that ∠SQL=50⁰ and ∠SRM=60⁰.
Here, the radius is OR = OQ.
We know that tangents drawn from an external point are perpendicular to the radius at the point of contact.
So, ∠OQL= 90 °  .
∠OQL=∠SQL+∠OQS  
∠OQS=∠OQL−∠SQL  
∠OQS= 90 ° − 50 °   ⇒∠OQS= 40 °   As OQ=OS   (Radius of the same circle)
⇒∠OSQ=∠OQS= 40 °  
As the angle between the radius and the tangent is 90°, therefore,
∠ORM= 90 °   So,
∠ORM=∠SRM+∠ORS  
∠ORS=∠ORM−∠SRM  
∠ORS= 90 ° − 60 °   ⇒∠ORS= 30 °    As OR=OS   (Radius of the same circle)
⇒∠OSR=∠ORS= 30 °  
Then, the measure of ∠QSR is.
∠QSR=∠OSQ+∠OSR    ⇒∠QSR= 40 ° + 30 °   ⇒∠QSR= 70 °   Hence, the measure of ∠QSR is 70 ° .  
Therefore, option 4 is correct.
 
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