# Perpendicular Lines

Perpendicular lines have the property that the product of their slopes is −1.

Mathematically, we say if a line has
slope *m*_{1} and another line has slope
*m*_{2} then the lines are **perpendicular**
if

m_{1}×m_{2}=−1

In the example at right, the slopes of the lines are `2` and `-0.5` and we have:

2 × −0.5 = −1

So the lines are perpendicular.

## Opposite Reciprocal

Another way of finding the slope of a perpendicular line is to find the opposite reciprocal of the slope of the original line.

In plain English, this means turn the original slope upside down and take the negative.

## Interactive graph - perpendicular lines

You can explore the concept of perpendicular lines in the following JSXGraph (it's not a fixed image).

**Drag** any of the points A, B or C and observe the slopes *m*_{1}, *m*_{2} of the 2 perpendicular lines.

You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.

Sometimes the labels overlap. It can't be helped!

If you get lost, you can always refresh the page.

### Special Cases

What if one of the lines is parallel to the *y*-axis?

For example, the line *y* = 3 is parallel to the *x*-axis and has slope `0`. The line *x* = 3.6 is parallel to the *y*-axis and has an **undefined** slope.

The lines are clearly perpendicular, but we cannot find the product of their slopes. In such a case, we cannot draw a conclusion from the product of the slopes, but we can see immediately from the graph that the lines are perpendicular.

The same situation occurs with the ** x- and y-axes**. They are perpendicular, but we cannot calculate the product of the 2 slopes, since the slope of the

*y*-axis is undefined.

### Exercise 1

A line *L *has slope
`m = 4`.

a) What is the slope of a line **parallel**
to *L?*

b) What is the slope of a line
**perpendicular** to *L?*

Answer

a) Since parallel lines have the same slope, the slope will be `4`.

b) Using *m*_{1} × *m*_{2}*=* -1, with *m*_{1} = 4, we obtain the value for *m*_{2}:

`m_2=-1/(m_1)`

`=-1/4`

### Exercise 2

A line passes through (-3, 9) and (4, 4). Another line passes through (9, -1) and (4, -8). Are the lines parallel or perpendicular?

Answer

The line through `(-3,9)` and `(4,4)` has slope

`m_1=(4-9)/(4-(-3))=(-5)/7`

The line through `(9,-1)` and `(4,-8)` has slope

`m_2=(-8-(-1))/(4-9)=(-7)/(-5)=7/5`

Now

`m_1m_2=(-5)/7xx7/5=-1`

Since the product of the slopes is `−1`, we conclude the lines are **perpendicular**.

Note: We could have sketched the lines to determine whether they were parallel or perpendicular.