A triangle has vertices P, Q and R - Edexcel - GCSE Maths - Question 19 - 2017 - Paper 2

To find the coordinates of point N, the midpoint of line segment QR, we use the same midpoint formula:

$N = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Substituting the coordinates of Q and R:

$N = \left(\frac{1 + 5}{2}, \frac{4 + -2}{2}\right) = \left(3, 1\right)$

Next, we need to calculate the gradients of lines MN and PR. The gradient (slope) is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For MN, using the coordinates of points M and N:

$m_{MN} = \frac{1 - (-1)}{3 - (-1)} = \frac{2}{4} = \frac{1}{2}$

For PR, using the coordinates of points P and R:

$m_{PR} = \frac{-2 - (-6)}{5 - (-3)} = \frac{4}{8} = \frac{1}{2}$

Since the gradients of line segments MN and PR are equal, we can conclude that MN is parallel to PR:

$m_{MN} = m_{PR} = \frac{1}{2}$

Thus, we have shown that MN is parallel to PR.