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In figure, tangents PQ and PR are drawn ...

In figure, tangents PQ and PR are drawn to a circle such that `angleRPQ=30^(@)`. A chord RS is drawn parallel to the tangent PQ. Find the `angleRQS`.

A

`30^@`

B

`40^@`

C

`50^@`

D

`60^@`

Text Solution

Verified by Experts

The correct Answer is:
A
PQ and PR are two tangents drawn from an external point P.

`:.`PQ=PR
[the lengths of tangents drawn from an external point to a circle are equal]
`rArranglePQR=angleQRP`
[angles opposite to equal sides are equal]
Now, in `DeltaPQR" "anglePQR+angleQRP+angleRPQ=180^(@)`
[sum of all interior angles of any triangle is `180^(@)`]
`rArranglePQR+anglePQR+30^(@)=180^(@)`
`rArr2anglePQR=180^(@)-30^(@)`
`rArranglePQR=(180^(@)-30^(@))/(2)=75^(@)`
Since, SR/QP
`:.angleSRQ=angleRQP=75^(@)` [alternate interior angles]
Also, `:.anglePQR=angleQSR=75^(@)` [by alternate segment theorem]
In `DeltaQRS, angleQ+angleR+angleS=180^(@)`
[sum of all interior angles of any triangle is `180^(@)`]
`rArrangleQ=180^(@)-(75^(@)+75^(@))`
`=30^(@)`
`:.angleRQS=30^(@)`
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