JAMB 1985 · UME · Q43

Circle geometry

In the figure, $PQ$ is the tangent from $P$ to the circle $QRS$ , with $SR$ a diameter and $SRP$ a straight line. If $\angle PQR = \theta$ and $\angle QPR = \phi$ , which of the following relationships is correct?

Worked solution (try it first)

The angle between the tangent $PQ$ and the chord $QR$ equals the angle in the alternate segment, so $\angle QSR = \theta$ .
$SR$ is a diameter, so $\angle SQR = 90^\circ$ (angle in a semicircle).
The whole angle $SQP$ is $90^\circ + \theta$ .
The angles of triangle $SQP$ add up to $180^\circ$ : $\theta + (90^\circ + \theta) + \phi = 180^\circ$ .
So $\phi = 90^\circ - 2\theta$ , option B.

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