Geometry Circles – Angle in Same Segment

Q,R,S,T are four points on the circumference of a circle. QT is the perpendicular bisector of RS. PQ is a tangent drawn to the circle from an external point P. If angle(SQP) = 20, find angle(TRS).

Answer: 70 degrees

Consider the diagram below.

QT is the perpendicular bisector of a chord RS. Hence QT has to pass through the centre of the circle.

Angle(OQP) = 90 as the radius of a circle meets a tangent perpendicularly at the point of tangency.

Angle(SQP) = 20. Hence angle(SQO) = 70.

Now for chord TS, angle(TQS) = angle(TRS) as they are in the same segment.

Hence angle(TRS) = 70.