WAEC 2021 · Paper 2 · Q4✱✱

(a)
$\overline{RU}$ and $\overline{RS}$ are tangents to the circle with centre $O$ at $U$ and $S$ , and $T$ is a point on the major arc $US$ . If $\angle URS = 80^\circ$ , find $\angle UTS$ .
(b)
$P$ is a point $270\text{ m}$ from $Q$ on a bearing of $328^\circ$ and $R$ is $420\text{ m}$ from $Q$ on a bearing of $058^\circ$ . Find the bearing of $P$ from $R$ .

Worked solution (try it first)

A tangent is perpendicular to the radius, so $\angle OUR = \angle OSR = 90^\circ$ .
The angles of quadrilateral $OURS$ add up to $360^\circ$ : $\angle UOS = 360^\circ - 90^\circ - 90^\circ - 80^\circ$
$= 100^\circ$ .
$T$ is on the major arc, so $\angle UTS$ is half the angle at the centre: $\angle UTS = 50^\circ$ .

Draw north at $Q$ .
$P$ is 270 m on $328^\circ$ and $R$ is 420 m on $058^\circ$ .
The angle between them is $(360^\circ - 328^\circ) + 58^\circ = 90^\circ$ , so triangle $PQR$ is right-angled at $Q$ .
At $R$ : $\tan\angle QRP = \frac{270}{420}$ , so $\angle QRP \approx 32.7^\circ$ .
At $R$ , the direction back to $Q$ is $058^\circ + 180^\circ = 238^\circ$ , and $P$ is $32.7^\circ$ further round clockwise.
Bearing of $P$ from $R$ $= 238^\circ + 32.7^\circ$
$\approx 271^\circ$ .