In Fig. 7, 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. - Mathematics

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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.

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Solution

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°

 2PQR + 30° = 180°

2PQR = 180° 30°

∴  2PQR = 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°

  Is there an error in this question or solution?
Chapter 8: Circles - Exercise 8.2 [Page 37]

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