On a graph sheet, using a scale of 2cm to 2 units on both axes, (a) Draw the straight line joining points P(-5, 3) and Q(2, 3); (b) construct the locus L of...

Answer Details

Scale: 2 cm to 2 units on both axes, i.e. 1 unit = 1 cm. So a length of 5 cm on the paper is the same as 5 units on the graph, and a side of the required rhombus measures 5 units.

Notice first that P(-5, 3) and Q(2, 3) have the same y-coordinate (3), so PQ is a horizontal line of length \(|2-(-5)| = 7\) units = 7 cm.

(a) The straight line PQ

P(-5, 3) and Q(2, 3) are plotted and joined by a straight line. This is the horizontal segment shown in blue at the level \(y = 3\).

(b) Locus L of points equidistant from P and Q

The set of points equidistant from P and Q is the perpendicular bisector of PQ. It is constructed by opening the compass to more than half of PQ, drawing equal arcs centred on P and on Q above and below the line, and joining their points of intersection. This bisector cuts PQ at its midpoint

\[ M = \left(\frac{-5+2}{2},\; 3\right) = (-1.5,\; 3), \]

and runs vertically, so the locus is the line \(x = -1.5\), shown dashed and labelled L.

(c) Locating R and S so that PRQS is a rhombus of side 5 cm

In the rhombus PRQS the four equal sides are PR, RQ, QS and SP, each 5 cm. Hence each of R and S must be 5 cm from P and 5 cm from Q. Setting the compass to 5 cm (= 5 units) and drawing arcs centred on P and on Q, the arcs intersect on the locus L at two points: R above PQ and S below PQ. Because PR = RQ = QS = SP = 5 cm, the figure PRQS is a rhombus, and its diagonals PQ and RS cross at right angles at M.

(d) Results

(i) Coordinates of R and S. R and S lie on L, so each has \(x = -1.5\). Taking R\((-1.5,\,y)\) with PR = 5:

\[ \sqrt{(-1.5-(-5))^{2} + (y-3)^{2}} = 5 \]\[ 3.5^{2} + (y-3)^{2} = 5^{2} \;\Rightarrow\; (y-3)^{2} = 25 - 12.25 = 12.75 \]\[ y - 3 = \pm\sqrt{12.75} = \pm 3.55 \;\Rightarrow\; y = 6.5 \text{ or } y = -0.6. \]

Therefore, reading from the construction,

\[ \boxed{R(-1.5,\; 6.5) \quad\text{and}\quad S(-1.5,\; -0.6).} \]

(ii) Area of the rhombus. The diagonals of a rhombus bisect each other at right angles, so the area is half the product of the diagonals. Here the diagonals are PQ and RS:

\[ PQ = 7\ \text{cm}, \qquad RS = 6.5-(-0.6) = 7.1\ \text{cm}. \]\[ \text{Area} = \frac{1}{2}\times PQ \times RS = \frac{1}{2}\times 7 \times 7.1 = \frac{49.7}{2} \]\[ \boxed{\text{Area} \approx 24.85\ \text{cm}^{2}\ (\approx 25\ \text{cm}^{2}).} \]