Using ruler and a pair of compasses only, (a) construct a quadrilateral PXYQ such that /PX/ = 9.9 cm, /QX/ = 10.2 cm, < QPZ = 75°, /QY/ = 10.4 cm and PQ // ...

The accurate scale drawing of the whole construction is shown below. All construction arcs and the two loci are kept visible, and the required lengths and angle are transferred from the data:

(a) Constructing quadrilateral \(PXYQ\)

1. Draw a base line and mark \(PX = 9.9\) cm.
2. At \(P\), construct \(\angle QPX = 75^{\circ}\): first construct \(60^{\circ}\), then \(90^{\circ}\), and bisect the \(60^{\circ}\)-\(90^{\circ}\) interval to add \(15^{\circ}\), giving \(60^{\circ}+15^{\circ}=75^{\circ}\).
3. With centre \(X\) and radius \(10.2\) cm, draw an arc to cut the \(75^{\circ}\) arm at \(Q\), so that \(|QX| = 10.2\) cm.
4. Through \(X\), construct a line parallel to \(PQ\) (copy \(\angle QPX\) at \(X\) on the same side), giving the direction of \(XY\) with \(PQ \parallel XY\).
5. With centre \(Q\) and radius \(10.4\) cm, draw an arc to cut the parallel line at \(Y\), so that \(|QY| = 10.4\) cm.
6. Join \(XY\), \(YQ\) and \(QP\) to complete the quadrilateral \(PXYQ\).

(b) The two loci

(i) Locus \(l_{1}\) - points equidistant from \(X\) and \(Y\) form the perpendicular bisector of \(XY\). With centre \(X\) and then centre \(Y\), using the same radius (greater than \(\tfrac{1}{2}|XY|\)), draw arcs above and below \(XY\); the line through their intersections is \(l_{1}\).

(ii) Locus \(l_{2}\) - points equidistant from the lines \(QY\) and \(YX\) lie on the bisector of \(\angle QYX\). With centre \(Y\), draw an arc cutting \(YQ\) and \(YX\); from those two cuts, using equal radii, draw arcs to meet, and join \(Y\) to that meeting point to obtain \(l_{2}\).

(c) Locating \(M\)

\(M\) is the point where \(l_{1}\) and \(l_{2}\) cross, as marked on the diagram.

(d) Measuring \(|PM|\)

Joining \(P\) to \(M\) and measuring with the ruler:

\(|PM| = 9.7\text{ cm}\)