P and Q are the points (−2, 6) and (10, 0) - Scottish Highers Maths - Question 2 - 2023

The gradient (slope) of the line segment PQ is calculated using the formula:

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

For points P(−2, 6) and Q(10, 0):

$m_{PQ} = \frac{0 - 6}{10 - (-2)} = \frac{-6}{12} = -\frac{1}{2}$

The gradient of the perpendicular bisector is the negative reciprocal of the gradient of PQ. Therefore:

$m_{perpendicular} = -\frac{1}{m_{PQ}} = -\frac{1}{-\frac{1}{2}} = 2$

To find the equation of the perpendicular bisector line, we can use the point-slope form of the line equation:

$y - y_1 = m(x - x_1)$

Using the midpoint (4, 3) and the perpendicular gradient 2:

$y - 3 = 2(x - 4)$

Expanding this gives:

$y - 3 = 2x - 8$ $y = 2x - 5$

Thus, the equation of the perpendicular bisector is:

$y = 2x - 5$