Three towns X, Y and Z are such that Y is 20 km from X and 22 km from Z. Town X is 18 km from Z. A health centre is to be built by the government to serve t...

(a) Construction using scale \(1\text{ cm}:2\text{ km}\)

1. Draw \(XZ=18\) km, which is \(9\) cm on the scale.
2. With centre \(X\), draw an arc of radius \(10\) cm. With centre \(Z\), draw an arc of radius \(11\) cm. Their intersection gives town \(Y\), so that \(XY=20\) km and \(YZ=22\) km.
3. Construct the perpendicular bisector of \(XY\). Every point on this line is the same distance from \(X\) and \(Y\).
4. With centre \(Z\), draw a circle of radius \(5\) cm, representing \(10\) km.
5. The two points where this circle crosses the perpendicular bisector are the possible health-centre locations.

(b) There are two possible locations, because the perpendicular bisector of \(XY\) intersects the circle centred at \(Z\) at two points.

(c) Measuring from \(X\) on an accurate scale construction gives approximately:

• \(XH_1 \approx 12.7\) km;
• \(XH_2 \approx 28.0\) km.

Values such as \(12.4\) km and \(27.6\) km can result from ruler measurement on a less precise drawing, but the intended readings are approximately \(12\)–\(13\) km and \(28\) km respectively.

(d) The convenient location is \(H_1\), the site approximately \(12.7\) km from \(X\). It lies inside the triangle formed by the three towns, while still being \(10\) km from \(Z\) and equidistant from \(X\) and \(Y\). The other site lies well outside the triangle and is much farther from \(X\) and \(Y\).

Examination reminder: “Equal distance from two towns” means construct the perpendicular bisector of the line joining them. “A fixed distance from a town” means draw a circle centred on that town.