Question

Write the equation of the perpendicular bisector of the segment AB.
A (1, 5) and B (7, -1)

Answer 1

The equation of the perpendicular bisector of the segment AB is
`y = x - 2`

Explanation:

Equation of a line:

The equation of a line has the following format:

`y = mx + b`

In which m is the slope and b is the y-intercept.

Perpendicular lines and slopes:

If two lines are perpendicular, the multiplication of their slopes is -1.

Equation of the perpendicular bisector of the segment AB.

Equation of a line that passes through the midpoint of segment AB and is perpendicular to AB.

Midpoint of segment AB:

Mean of their coordinates x and y. So

(1+7)/2 = 4

(5-1)/2 = 2

So (4,2).

Slope of segment AB:

When we have two points, the slope between them is given by the change in y divided by the change in x.

In segment AB, we have points (1,5) and (7,-1). So

Change in y: -1 - 5 = -6

Change in x: 7 - 1 = 6

Slope: -6/6 = -1

Equation of the perpendicular bisector:

The slope, multiplied with the slope of segment AB, is -1. So

`-1m = -1`

`m = 1`

So

`y = x + b`

Passes through (4,2), which means that when
`x = 4, y = 2`. So

`y = x + b`

`2 = 4 + b`

`b = -2`

So

`y = x - 2`