WAEC 2023 · Paper 2 · Q12

(a)
In the diagram, $P$ , $Q$ , $R$ and $S$ are points on the circle with centre $O$ ; $QR \parallel OS$ , $\angle QOR = 2m$ , $\angle QPR = n$ and $\angle SOR = 54^\circ$ . Find the values of $m$ and $n$ .
Separate values with commas, e.g. 3, −2
(b)
The length of a rectangle is $4\text{ cm}$ more than its width. If the perimeter is $40\text{ cm}$ , find its area.

Worked solution (try it first)

$QR \parallel OS$ , so $\angle ORQ = \angle SOR = 54^\circ$ (alternate angles).
$|OQ| = |OR|$ (radii), so triangle $OQR$ is isosceles and $\angle OQR = \angle ORQ = 54^\circ$ .
Angles in triangle $OQR$ : $\angle QOR = 180^\circ - 54^\circ - 54^\circ$
$= 72^\circ$ .
So $2m = 72^\circ$ and $m = 36^\circ$ .
The angle at the centre is twice the angle at the circumference on the same arc $QR$ , so $n = \frac12 \times 72^\circ = 36^\circ$ .