Question

In figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS.

[Hint: Draw a line through Q and perpendicular to QP.]

Answer 1

PQ and PR are two tangents drawn from an external point P.

∴ PQ = PR  ...[The length of tangents drawn from an external point to a circle are equal]

⇒ ∠PQR = ∠QRP  ...[Angles opposite to equal sides are equal]

Now, In ΔPQR,  

∠PQR + ∠QRP + ∠RPQ = 180°  ...[Sum of all interior angles of any triangle is 180°]

⇒ ∠PQR + ∠PQR + 30° = 180°

⇒ 2∠PQR = 180° – 30° = 150°  ...[∵ ∠PQR = ∠QRP]

⇒ ∠PQR = `(180^circ - 30^circ)/2` = 75°

Since, SR || QP

∴ ∠SRQ = ∠RQP = 75°   ...[Alternative interior angles]

Also, ∠PQR = ∠QSR = 75°  ...[By alternative segment theorem]

In ΔQRS,

∠Q + ∠R + ∠S = 180°  ...[Sum of all interior angles of any triangles is 180°]

⇒ ∠Q = 180° – (75° + 75°) = 30°

∴ ∠RQS = 30°