The diagram below shows a map of Ireland on a co-ordinate grid - Junior Cycle Mathematics - Question 9 - 2017

To draw the line segment, connect points $B(3, 1)$ and $H(8, 5)$ with a straight line on the grid.

1. Place the compass on point $B$ and draw an arc above and below the line segment.
2. Without changing the compass width, repeat the process from point $H$, creating two intersection points.
3. Use a straight edge to connect these two intersection points, forming the perpendicular bisector.

To find the midpoint, use the formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ where $(x_1, y_1) = (3, 1)$ and $(x_2, y_2) = (8, 5)$.

Calculating: $M = \left( \frac{3 + 8}{2}, \frac{1 + 5}{2} \right) = \left( \frac{11}{2}, \frac{6}{2} \right) = \left( 5.5, 3 \right)$

Thus, Midpoint = $(5.5, 3)$.

Use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting $(x_1, y_1) = (8, 4)$ and $(x_2, y_2) = (3, 6)$: $d = \sqrt{(3 - 8)^2 + (6 - 4)^2} = \sqrt{(-5)^2 + (2)^2} = \sqrt{25 + 4} = \sqrt{29}$

Thus, the distance is $\sqrt{29}$ cm.

Given that 4 cm on the diagram corresponds to 180 km, the scale can be calculated as: $\text{Kilometers per cm} = \frac{180 \text{ km}}{4 \text{ cm}} = 45 \text{ km/cm}$