Step by step
The question
Question: find the perpendicular bisector of A(2, −5) and B(8, 3).
The perpendicular bisector cuts A B in half, at a right angle.
So it goes through the middle of A B.
1 Find the midpoint
The midpoint is the average of the x's and the average of the y's: (5, −1).
2 Slope of AB
The top is the change in y: second y minus first y.
The bottom is the change in x: second x minus first x.
That is 8/6, so 4/3.
3 Perpendicular slope
Perpendicular slopes: flip the fraction and change the sign.
4/3 becomes −3/4.
Check: 4/3 times −3/4 is negative 1, as for every perpendicular pair.
Careful!
The slope of A B itself would give a line along A B, not across it.
We need the perpendicular slope.
4 Write the equation
The line is y = m x + b with m = −3/4.
It goes through M: put x = 5 and y = −1 in.
−3/4 times 5 is −15/4.
To get b alone, add 15/4 on both sides: b = 11/4.
So the perpendicular bisector is y = −3/4, times x, plus 11/4.
5 Draw it
Let's draw it on the grid.
A and B, and the segment between them.
M in the middle.
The bisector goes through M, across A B.
It meets A B at a right angle.
6 Check
Let's check with another point on the line, (9, −4).
Its distance squared to A and to B is the same: 50.
Every point on the bisector is as far from A as from B.
So it is right!
Let's look back.
First: find the midpoint.
Then: slope of AB.
Then: perpendicular slope.
Then: write the equation.
Then: draw it.
Last: check.