Question

In the given figure, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS.

In the given 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 measure of ∠RQS.

Answer 1

It is given that, ∠RPQ = 30° and PQ and PR are tangents drawn to a circle from P to the same circle.

∴ PQ = PR          ...(Tangents drawn from an external point to a circle are equal in length.)

In ∆PQR,

PQ = PR

∴ ∠PQR = ∠PRQ        ...(Angles opposite to equal sides are equal.)

Now, In ∆PQR,

∠PQR + ∠PRQ + ∠RPQ = 180°   ...(Angle sum property of a triangle)

∴ ∠PQR + ∠PQR + 30° = 180°

∴  2∠PQR + 30° = 180°

∴ 2∠PQR = 180° − 30°

∴  2∠PQR = 150°

∴ ∠PQR = 75°

So, ∠PQR = ∠QRS = 75°     ...(Alternate angles)

∠PQR = ∠QSR = 75°       ...(Alternate segment angles are equal)

and ∠QRS = ∠QSR = 75°

∴ ΔQRS is also an isosceles triangle.

∴ ∠QRS + ∠QSR + ∠RQS = 180°

∴ 75° + 75° + ∠RQS = 180°

∴ ∠RQS = 180° − 150°

∴ ∠RQS = 30°