WAEC 2009 · Paper 2 · Q10

(a)
Using a ruler and a pair of compasses only, construct: (i) quadrilateral $PQRS$ such that $|PQ| = 10$ cm, $|QR| = 8$ cm, $|PS| = 6$ cm, $\angle PQR = 60^\circ$ and $\angle QPS = 75^\circ$ ; (ii) the locus $l_1$ of points equidistant from $QR$ and $RS$ ; (iii) the locus $l_2$ of points equidistant from $R$ and $S$ .
Model answer
Draw $PQ = 10$ cm. At $Q$ construct $60^\circ$ and mark $R$ with $QR = 8$ cm. At $P$ construct $75^\circ$ ( $60^\circ$ plus half of the $30^\circ$ between $60^\circ$ and $90^\circ$ ) and mark $S$ with $PS = 6$ cm. Join $RS$ . $l_1$ is the bisector of angle $QRS$ ; $l_2$ is the perpendicular bisector of $RS$ . Leave every construction arc visible.
(b)
Measure $|RS|$ .

Worked solution (try it first)

Draw $PQ = 10$ cm.
At $Q$ , construct $60^\circ$ (an arc, then the same radius from where it cuts $QP$ ) and mark $R$ on the arm with $QR = 8$ cm.
At $P$ , construct $90^\circ$ and $60^\circ$ , then bisect the angle between them to get $75^\circ$ .
Mark $S$ with $PS = 6$ cm and join $RS$ .

Points equidistant from the lines $QR$ and $RS$ lie on the bisector of angle $QRS$ : draw it and label it $l_1$ .

Points equidistant from $R$ and $S$ lie on the perpendicular bisector of $RS$ : draw it and label it $l_2$ .