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In the figure, tangents \( P Q \) and \( P R \) are drawn from an external point \( P \) to a circle with centre $O$, such that \( \angle R P Q=30^{\circ} . \) A chord \( R S \) is drawn parallel to the tangent \( P Q \). Find \( \angle R Q S \).
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Given:
In the figure, tangents \( P Q \) and \( P R \) are drawn from an external point \( P \) to a circle with centre $O$, such that \( \angle R P Q=30^{\circ} . \) A chord \( R S \) is drawn parallel to the tangent \( P Q \).
To do:
We have to find \( \angle R Q S \).
Solution:
$PQ$ and $PR$ are tangents to the circle with centre $O$ drawn from $P$ and $\angle RPQ = 30^o$
$RS\ \parallel\ PQ$.
$PQ = PR$ (Tangents to the circle from an external point are equal)
$\angle PRQ = \angle PQR$
$\angle PRQ=\angle PQR=\frac{180^{\circ}-30^{\circ}}{2}$
$=\frac{150^{\circ}}{2}=75^{\circ}$
$\angle SRQ=\angle PQR=75^{\circ}$
$\angle RSQ=\angle QRS$ (since $QR=QS$)
$=75^{\circ}$
In $\Delta QRS$,
$\angle RQS=180^{\circ}-(\angle RSQ+\angle QRS)$
$=180^{\circ}-(75^{\circ}+75^{\circ})$
$=180^{\circ}-150^{\circ}$
$=30^{\circ}$
Therefore, $\angle RQS=30^{\circ}$.