JAMB 1986 · UME · Q37

$PQ$ and $PR$ are tangents from $P$ to a circle with centre $O$ . If $\angle QRP = 34^\circ$ , find the angle marked $x$ ( $\angle QOR$ ).

Worked solution (try it first)

$\angle QRP$ is the angle between the tangent $RP$ and the chord $RQ$ .
It equals the angle that $QR$ makes at any point on the major arc.
The angle at the centre is twice the angle at the circumference, so $x = 2 \times 34^\circ = 68^\circ$ , option C.
Check: tangents from $P$ are equal, so $\angle PQR = 34^\circ$ and $\angle QPR = 112^\circ$ .
In $OQPR$ the radii meet the tangents at $90^\circ$ , so $x = 360^\circ - 90^\circ - 90^\circ - 112^\circ$
$= 68^\circ$ ✓.