WAEC 2019 · Paper 2 · Q5✱✱

(a)
In the diagram, $MNPR$ is a circle with centre $O$ . The reflex angle at $O$ is $196^\circ$ and $\angle NMO = 52^\circ$ . Find the value of $m$ ( $= \angle OPN$ ).
(b)
A farmer uses $\frac25$ of his land to grow cassava, $\frac13$ of the remainder for plantain and the rest for yam. Find the part of the land used for yam.

Worked solution (try it first)

The reflex angle at $O$ is $196^\circ$ , so the angle $MOP$ on the other side is $360^\circ - 196^\circ = 164^\circ$ .
The angle at the circumference is half the angle at the centre: $\angle MNP = 82^\circ$ .
$OM = OP$ (radii), so triangle $OMP$ is isosceles: $\angle OMP = \angle OPM$
$= \frac{180^\circ - 164^\circ}{2}$
$= 8^\circ$ .
In triangle $MNP$ : $\angle NMP = 52^\circ + 8^\circ$
$= 60^\circ$ and $\angle NPM = m + 8^\circ$ .
So $60^\circ + 82^\circ + m + 8^\circ = 180^\circ$ , and $m = 30^\circ$ .