Work out the equation of the perpendicular bisector of the line from… Step by Step

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The question

Question: find the perpendicular bisector of A(2, 0) and B(6, 4).

perpendicular bisector of A(2, 0) and B(6, 4)

The perpendicular bisector cuts A B in half, at a right angle.

So it goes through the middle of A B.

bisector = cuts AB in half

perpendicular = at 90° to AB

1 Find the midpoint

The middle of two numbers: add them, then divide by 2.

We do it for the x's and for the y's.

M = (2 + 62, 0 + 42)

2 plus 6 is 8.

0 plus 4 is 4.

M = (82, 42)

8 divided by 2 is 4.

4 divided by 2 is 2.

So the midpoint is M = (4, 2).

M = (4, 2)

2 Slope of AB

The slope tells how steep A B is: how far up for each step across.

The top is the change in y: second y minus first y.

The bottom is the change in x: second x minus first x.

4 minus 0 is 4.

6 minus 2 is 4.

That is 4/4, so 1.

slope of AB = 4 − 06 − 2 = 44 = 1

3 Perpendicular slope

We need a line at a right angle to A B.

Turn A B a quarter turn: its steps right 4, up 4 turn into up 4, left 4.

AB: right 4, up 4

turned 90°: up 4, left 4

Its slope: up over across, 4 over −4, so −1.

The two numbers swapped places, and the sign changed.

slope = 4−4

= −1

So for a right angle: flip the slope and change its sign.

1 becomes −1.

perpendicular slope = −1

Careful!

The slope of A B itself would give a line along A B, not across it.

We need the perpendicular slope.

Careful: not slope 1 ✗

4 Write the equation

A straight line is y = m x + b.

m is the slope: here m = −1.

b is where it crosses the y-axis.

We find b.

y = mx + b

The line goes through M, so the numbers of M fit the equation.

Put x = 4 and y = 2 in.

2 = −1 × 4 + b

−1 times 4 is −4.

2 = −4 + b

Is b alone now?

Not yet: − 4 is still with b.

How do we remove it?

We want b by itself.

The −4 is in the way.

To remove −4, we add 4.

We do it on both sides, so they stay equal.

2 + 4 = −4 + b ~~+ 4~~

−4 and + 4 cancel each other.

They make zero.

2 plus 4 is 6.

So b = 6.

b = 6

So the perpendicular bisector is y = −x + 6.

y = −x + 6

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.

Every point on the bisector must be as far from A as from B.

Take another point on the line.

From M, go right 1, down 1: the slope's steps.

We land on (5, 1).

How far is it from A?

Use a² + b² = c².

Across: 5 minus 2 is 3.

Up or down: 1 minus 0 is 1.

3 squared plus 1 squared is 10.

to A: (5 − 2)² + (1 − 0)² = 3² + 1² = 10

To B: 6 minus 5 is 1.

4 minus 1 is 3.

1 squared plus 3 squared is 10.

The same!

So (5, 1) is as far from A as from B.

Right!

to B: (6 − 5)² + (4 − 1)² = 1² + 3² = 10

Let's look back.

First: find the midpoint.

Then: slope of AB.

Then: perpendicular slope.

Then: write the equation.

Then: draw it.

Last: check.

Answer