The sketch shows a plot of land . (a) Using a scale of 1 cm to 10m, draw an accurate diagram of the plot ; (b) Construct : (i) The locus \(l_{1}\) of points...

Answer Details

Solution

The completed scale construction is shown below. Everything that follows is read directly from it.

(a) Accurate scale drawing (1 cm to 10 m). Each real length is divided by 10 to get the drawing length:

\(|AB| = 85\text{ m} = \dfrac{85}{10} = 8.5\text{ cm}, \qquad |CA| = |CB| = 111\text{ m} = \dfrac{111}{10} = 11.1\text{ cm}.\)

Points A, B and C are plotted with ruler and compasses to these lengths, giving the triangular plot above.

(b) The two loci.

(i) \(l_1\), the locus of points equidistant from the sides \(AC\) and \(BC\), is the bisector of angle \(ACB\). It is constructed from C (shown by the two equal angle marks) and drawn as the blue dashed line.

(ii) \(l_2\), the locus of points 60 m from A, is a circle centred at A with radius \(\dfrac{60}{10} = 6\text{ cm}\). The relevant part of this arc is drawn in red.

(c) Locating T and finding \(|TC|\). The tree T lies on both loci, so it is the point where \(l_1\) meets \(l_2\) inside the plot. Measuring the drawing length of \(CT\):

\(CT = 6.0\text{ cm}.\)

Converting back to the real distance using the scale (multiply by 10):

\(|TC| = 6.0 \times 10 = 60\text{ m}.\)

(d) Region for the flagpole P. P must be:

• nearer AC than BC – this is the side of the bisector \(l_1\) lying toward \(AC\) (the upper part of the plot), and
• more than 60 m from A – this is the region outside the arc \(l_2\), i.e. on the C-side of the arc.

The set of points satisfying both conditions is the green hatched region above (bounded by side \(CA\), the bisector \(l_1\) and the arc \(l_2\)). Any point P chosen inside that shaded region is a valid position for the flagpole.