The co-ordinate diagram below shows part of the town where Ben lives - Junior Cycle Mathematics - Question 10 - 2019

The coordinates of Home are (-3, 5) and the Shop are (-3, -1).

Using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

We find:

$d = \sqrt{((-3) - (-3))^2 + ((-1) - (5))^2} = \sqrt{0 + (-6)^2} = \sqrt{36} = 6$

Thus, the distance from Home to the Shop is 6.0 cm.

Given that the distance on the grid is approximately 6 cm (from previous calculation), the actual distance can be calculated as follows:

Actual distance = diagram distance × scale

$\text{Actual distance} = 6 \times 25000 = 150000 \text{ cm}$

To convert this to meters:

$\text{Distance in meters} = \frac{150000}{100} = 1500 \text{ m}$

The slope (m) is calculated using:

$m = \frac{rise}{run}$

From Shop (-3, -1) to Home (-3, 5), the rise is (5 - (-1) = 6) and the run is ((-3 - (-3)) = 0). This indicates that the run measurement would be taken from different coordinates:

With coordinates corrected from (-3, -1) to (0, -1):

The rise remains 6 and the run becomes 18 - (-3) = 3, hence,

The slope is:

$m = \frac{6}{18} = \frac{1}{3}$

To find the angle ( H ), we use the inverse tangent function:

$H = \tan^{-1}\left(\frac{1}{3}\right)$

Using a calculator, we determine:

$H \approx 18.43°$

Thus, rounding to the nearest degree, ( H = 18°).