Step by step
The question
Question: find the perpendicular bisector of A(1, 5) and B(5, 1).
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, 3).
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 −4/4, so −1.
3 Perpendicular slope
Perpendicular slopes: flip the fraction and change the sign.
−1 becomes 1.
Check: −1 times 1 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.
It goes through M: put x = 3 and y = 3 in.
1 times 3 is 3.
To get b alone, take away 3 on both sides: b = 0.
So the perpendicular bisector is y = x.
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, (4, 4).
Its distance squared to A and to B is the same: 10.
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.