Step by step
The question
Question: find the perpendicular bisector of A(−6, 1) and B(2, 7).
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: (−2, 4).
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 6/8, so 3/4.
3 Perpendicular slope
Perpendicular slopes: flip the fraction and change the sign.
3/4 becomes −4/3.
Check: 3/4 times −4/3 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 = −4/3.
It goes through M: put x = −2 and y = 4 in.
−4/3 times −2 is 8/3.
To get b alone, take away 8/3 on both sides: b = 4/3.
So the perpendicular bisector is y = −4/3, times x, plus 4/3.
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, (1, 0).
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.