(a) Using ruler and a pair of compasses only, construct : (i) quadrilateral PQRS such that /PQ/ = 10 cm, /QR/ = 8 cm, /PS/ = 6 cm, < PQR = 60° and < QPS = 7...
Question 1 Report
(a) Using ruler and a pair of compasses only, construct : (i) quadrilateral PQRS such that /PQ/ = 10 cm, /QR/ = 8 cm, /PS/ = 6 cm, < PQR = 60° and < QPS = 75° ;
(ii) the locus \(l_{1}\) of points equidistant from QR and RS ; (iii) locus \(l_{2}\) of points equidistant from R and S ;
(b) Measure /RS/.
Answer Details
Construction and loci
- Draw \(PQ=10\text{ cm}\).
- At \(Q\), construct an angle of \(60^\circ\) with \(QP\). On this ray, mark \(R\) so that \(QR=8\text{ cm}\).
- At \(P\), construct an angle of \(75^\circ\) with \(PQ\). This can be made by constructing \(60^\circ\) and \(90^\circ\), then bisecting the angle between them. On this ray, mark \(S\) so that \(PS=6\text{ cm}\).
- Join \(R\) to \(S\) to complete quadrilateral \(PQRS\).
- The locus \(l_1\), equidistant from \(QR\) and \(RS\), is the internal angle bisector of \(\angle QRS\).
- The locus \(l_2\), equidistant from points \(R\) and \(S\), is the perpendicular bisector of \(RS\).
Measurement of \(RS\): \(RS\approx 4.6\text{ cm}\), measured to the nearest \(0.1\text{ cm}\).
A check from the given lengths and angles gives \[ RS=\sqrt{(10-8\cos60^\circ-6\cos75^\circ)^2+(8\sin60^\circ-6\sin75^\circ)^2} \approx 4.59\text{ cm}. \] Therefore, \(4.6\text{ cm}\) is the appropriate measured value. The stated value \(4.4\text{ cm}\) is not consistent with an accurate construction using the dimensions given.
Examination reminder: “Equidistant from two intersecting sides” indicates an angle bisector; “equidistant from two points” indicates a perpendicular bisector.
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