Step by step
The question
Question: find the perpendicular bisector of A(1, −3) and B(5, 5).
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: (3, 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 8/4, so 2.
3 Perpendicular slope
Perpendicular slopes: flip the fraction and change the sign.
2 becomes −1/2.
Check: 2 times −1/2 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 = −1/2.
It goes through M: put x = 3 and y = 1 in.
−1/2 times 3 is −3/2.
To get b alone, add 3/2 on both sides: b = 5/2.
So the perpendicular bisector is y = −1/2, times x, plus 5/2.
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, (5, 0).
Its distance squared to A and to B is the same: 25.
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.