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.
R O P Q S 60°


Answer:

60

Step by Step Explanation:
  1. Let us join OQ and OR. Also, produce PQ and PR to M and N respectively.
    R O N M P Q S 60°
  2. 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=180ROQ=180RPQ=18060=120
  3. 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=12ROQ=12×120=60 As, RS//PQ, SQM=RSQ=60 [Alternate Interior Angles]  Also, PQR=RSQ=60 [Alternate Segment Theorem] 
  4. 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
  5. Thus, the measure of RQS is 60.

You can reuse this answer
Creative Commons License