(a) Using ruler and a pair of compasses only, construct a : (i) Trapezium WXYZ such that |WX| = 8 cm, |XY| = 5.5 cm, |YZ| = 8.3 cm, < WXY = 60° and WX // ZY...

Construction and measurements

(a)(i) Constructing \(WXYZ\)

1. Draw \(WX=8\text{ cm}\).
2. At \(X\), construct an angle of \(60^\circ\) with \(XW\). On this ray, mark \(Y\) so that \(XY=5.5\text{ cm}\).
3. Construct a line through \(Y\) parallel to \(WX\): construct a perpendicular to \(WX\), then construct a perpendicular to that new line through \(Y\).
4. On this parallel line, on the side towards \(W\), mark \(Z\) such that \(YZ=8.3\text{ cm}\). Join \(Z\) to \(W\).

(a)(ii) Constructing rectangle \(PQYZ\)

Construct perpendiculars from \(Y\) and \(Z\) to \(WX\). Their intersections with \(WX\) are \(Q\) and \(P\), respectively. Since \(PQ\) lies on \(WX\), \(PQ\parallel YZ\), and the other two sides are perpendicular to \(WX\), \(PQYZ\) is a rectangle.

(b)(i) Length \(QX\)

\(Q\) is the perpendicular projection of \(Y\) onto \(WX\). Therefore \(QX\) is the horizontal component of \(XY\):

\[ QX=XY\cos60^\circ=5.5\times0.5=2.75\text{ cm}. \]

Measured to one decimal place, \(QX\approx\mathbf{2.8\text{ cm}}\). A ruler measurement of \(2.7\text{ cm}\) may result from drawing accuracy or truncation.

(b)(ii) Angle \(XWZ\)

The height of the trapezium is

\[ YQ=5.5\sin60^\circ\approx4.763\text{ cm}. \]

Also, \(WQ=8-2.75=5.25\text{ cm}\), whereas \(PQ=YZ=8.3\text{ cm}\). Thus \(P\), and therefore \(Z\), is \(8.3-5.25=3.05\text{ cm}\) to the left of \(W\).

The acute angle between \(WZ\) and the leftward extension of \(WX\) is

\[ \tan^{-1}\left(\frac{4.763}{3.05}\right)\approx57.4^\circ. \]

However, \(\angle XWZ\) is measured from the ray \(WX\), which points to the right. Hence

\[ \angle XWZ=180^\circ-57.4^\circ\approx\mathbf{122.6^\circ}, \]

so the measured answer is about \(\mathbf{123^\circ}\).

The stated value \(75^\circ\) is not consistent with the given side lengths, the \(60^\circ\) angle at \(X\), and \(WX\parallel ZY\). A useful check is that \(Z\) lies to the left of \(W\), so \(\angle XWZ\) must be obtuse, not \(75^\circ\).