Worked examples
The question
Question: find the perpendicular bisector of A(1, 2) and B(5, 6).
perpendicular bisector of A(1, 2) and B(5, 6)
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 midpoint is the average of the x's and the average of the y's: (3, 4).
M = (1 + 52, 2 + 62) = (3, 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 4/4, so 1.
slope of AB = 6 − 25 − 1 = 44 = 1
3 Perpendicular slope
Perpendicular slopes: flip the fraction and change the sign.
1 becomes −1.
perpendicular slope = −1
Check: 1 times −1 is negative 1, as for every perpendicular pair.
1 → flip, change sign → −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
The line is y = m x + b with m = −1.
It goes through M: put x = 3 and y = 4 in.
4 = −1 × 3 + b
−1 times 3 is −3.
4 = −3 + b
To get b alone, add 3 on both sides: b = 7.
b = 7
So the perpendicular bisector is y = −x + 7.
= y = −x + 7
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, 3).
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!
point (4, 3) on the line: to A² = 10, to B² = 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.
▶ Watch it solved
The question
Question: find the perpendicular bisector of A(2, 3) and B(6, 1).
perpendicular bisector of A(2, 3) and B(6, 1)
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 midpoint is the average of the x's and the average of the y's: (4, 2).
M = (2 + 62, 3 + 12) = (4, 2)
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 −2/4, so −1/2.
slope of AB = 1 − 36 − 2 = −24 = −12
3 Perpendicular slope
Perpendicular slopes: flip the fraction and change the sign.
−1/2 becomes 2.
perpendicular slope = 2
Check: −1/2 times 2 is negative 1, as for every perpendicular pair.
−12 → flip, change sign → 2
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 −12 ✗
4 Write the equation
The line is y = m x + b with m = 2.
It goes through M: put x = 4 and y = 2 in.
2 = 2 × 4 + b
2 times 4 is 8.
2 = 8 + b
To get b alone, take away 8 on both sides: b = −6.
b = −6
So the perpendicular bisector is y = 2x − 6.
= y = 2x − 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 with another point on the line, (5, 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!
point (5, 4) on the line: to A² = 10, to B² = 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.
▶ Watch it solved
The question
Question: find the perpendicular bisector of A(−3, 8) and B(9, −2).
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.
bisector = cuts AB in half
perpendicular = at 90° to AB
1 Find the midpoint
The midpoint is the average of the x's and the average of the y's: (3, 3).
M = (−3 + 92, 8 + (−2)2) = (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.
slope of AB = −2 − 89 − (−3) = −1012 = −56
3 Perpendicular slope
Perpendicular slopes: flip the fraction and change the sign.
−5/6 becomes 6/5.
perpendicular slope = 65
Check: −5/6 times 6/5 is negative 1, as for every perpendicular pair.
−56 → flip, change sign → 65
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 −56 ✗
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.
3 = 65 × 3 + b
6/5 times 3 is 18/5.
3 = 185 + b
To get b alone, take away 18/5 on both sides: b = −3/5.
b = −35
So the perpendicular bisector is y = 6/5, times x, minus 3/5.
= y = 65x − 35
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!
point (8, 9) on the line: to A² = 122, to B² = 122
Let's look back.
First: find the midpoint.
Then: slope of AB.
Then: perpendicular slope.
Then: write the equation.
Then: draw it.
Last: check.
▶ Watch it solved
