JAMB 2010 · UTME · Q28

Circle geometry

In the diagram, $O$ is the centre of the circle, $ROS$ is a straight line produced to $U$ , $UT$ is a tangent to the circle at $T$ and $\angle TRS = 25^\circ$ . Find $x$ .

Worked solution (try it first)

$RS$ is a diameter, so $\angle RTS = 90^\circ$ (angle in a semicircle).
The angles of triangle $RTS$ add up to $180^\circ$ : $\angle TSR = 180^\circ - 90^\circ - 25^\circ$
$= 65^\circ$ .
$x$ is between the tangent and the chord $TR$ , so it equals the angle in the alternate segment: $x = \angle TSR = 65^\circ$ , option A.

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