Step by step
The question
Question: find the perpendicular bisector of A(−3, 8) and B(9, −2).
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 −10/12, so −5/6.
3 Perpendicular slope
Perpendicular slopes: flip the fraction and change the sign.
−5/6 becomes 6/5.
Check: −5/6 times 6/5 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 = 6/5.
It goes through M: put x = 3 and y = 3 in.
6/5 times 3 is 18/5.
To get b alone, take away 18/5 on both sides: b = −3/5.
So the perpendicular bisector is y = 6/5, times x, minus 3/5.
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, (8, 9).
Its distance squared to A and to B is the same: 122.
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.