In the given figure, tangents PQ and PR are drawn from an external point P to a circle with center O, such that ∠RPQ=60∘. A chord RS is drawn parallel to the tangent PQ. Find the measure of ∠RQS.
Answer:
60∘
- Let us join OQ and OR. Also, produce PQ and PR to M and N respectively.
- We know that the angle between two tangents from an external point is supplementary to the angle subtended by the radii at the center.
Thus, ∠RPQ+∠ROQ=180∘⟹∠ROQ=180∘−∠RPQ=180∘−60∘=120∘ - We also know that the angle subtended by an arc at the center is twice the angle subtended by the same arc on the remaining part of the circle.
So, ∠RSQ=12∠ROQ=12×120∘=60∘ As, RS//PQ, ⟹∠SQM=∠RSQ=60∘ [Alternate Interior Angles] Also, ∠PQR=∠RSQ=60∘ [Alternate Segment Theorem] - We know that the sum of angles on a straight line is 180∘.
As PM is a straight line. ⟹∠SQM+∠RQS+∠PQR=180∘ Therefore, ∠RQS=180∘−(∠SQM+∠PQR)=180∘−(60∘+60∘)=60∘ - Thus, the measure of ∠RQS is 60∘.