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 $."

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, $\angle RQS=30^{\circ}$.