(a) ABCD is a trapezium in which AB // DC, |AB| = 8cm, < ABC = 60°, |BC| = 5.5cm and |BD| = 8.3cm. Using a ruler and a pair of compasses only, construct: (i...

There is an inconsistency in the supplied materials: the question asks for \(|AB|\), which is already given as \(8\text{ cm}\), whereas the reference answer gives \(|AC|=7\text{ cm}\). The reference answer therefore appears to intend \(|AC|\), not \(|AB|\).

Construction

1. Draw \(AB=8\text{ cm}\).
2. At \(B\), construct an angle of \(60^\circ\) with \(BA\). On this ray, mark \(C\) so that \(BC=5.5\text{ cm}\).
3. Construct a line through \(C\) parallel to \(AB\): construct a perpendicular to \(AB\) through \(C\), then construct a perpendicular to this new line through \(C\).
4. With centre \(B\) and radius \(8.3\text{ cm}\), draw an arc to meet the parallel through \(C\) at \(D\), choosing the intersection that gives the trapezium shown. Join \(A\) to \(D\) and \(C\) to \(D\).
5. Construct perpendiculars from \(D\) and \(C\) to \(AB\). Let their feet on \(AB\) be \(P\) and \(Q\), respectively. Since \(PQ\parallel DC\) and \(PD\parallel QC\), with right angles at \(P\) and \(Q\), \(PQCD\) is a rectangle.

Measurements

• \(|AB|=8.0\text{ cm}\), because this length was given and used to construct the figure.
• \(|QB|\approx2.8\text{ cm}\), which may be recorded as \(2.9\text{ cm}\) from a construction drawing.

The value of \(QB\) follows from the horizontal component of \(BC\): \(QB=5.5\cos60^\circ=2.75\text{ cm}\), so a ruler measurement should be close to \(2.8\text{ cm}\) or \(2.9\text{ cm}\).

If the intended measurement was \(|AC|\), as indicated by the reference answer, then \(|AC|\approx7.1\text{ cm}\), recorded as \(7\text{ cm}\).