Question

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

Answer is 30° but solve karke btao jaldi please....

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Answer 1

PQ = PR

Since tangents drawn from an external point to a circle are equal.

And PQR is an isosceles triangle

thus, ∠RQP = ∠QRP

∠RQP + ∠QRP + ∠RPQ = 180° [Angle sum property of a triangle]

2∠RQP + 30° = 180°

2∠RQP = 150°

∠RQP = ∠QRP = 75°

∠RQP = ∠RSQ = 75°  [ Angles in alternate Segment Theorem states that angle between chord and tangent is  equal to the angle in the alternate segment]

RS is parallel to PQ

Therefore ∠RQP = ∠SRQ = 75°    [Alternate angles]

∠RSQ = ∠SRQ = 75°

Therefore QRS is also an isosceles triangle

∠RSQ + ∠SRQ + ∠RQS = 180°  [Angle sum property of a triangle]

75° + 75° + ∠RQS = 180°

150° + ∠RQS = 180°

∠RQS = 30°