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`.

Options

A. `30^@`
B. `40^@`
C. `50^@`
D. `60^@`

Answer

A

Explanation

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^(@)`