Perpendicular Lines

These two lines are perpendicular: they meet at a 90° angle.

We could try to check whether or not these lines are really perpendicular by using a protractor, but it's hard to measure a picture on a computer monitor without making mistakes. Can we find out whether lines are perpendicular without measuring their angles?

Let's start by comparing the slope of these lines.

m = (y₂ - y₁) / (x₂ - x₁)

Blue line:

m = (2 - 4) / (1/2 - 0)

= -2/ (1/2)

= -4

Red line:

m = (1 - 1/2) / (4 - 2)

= (1/2)/2

= 1/4

The slope of the blue line is -4, and the slope of the red line is 1/4. Do these numbers have anything in common? You might notice that 1/(-4) = -1/4, the opposite of the red line's slope. Let's call the slope of the blue line m_blue and the slope of the red line m_red. Then we can write 1/(-4) = -1/4 as:

1 / m_blue = -m_red

If we multiply both sides of the equation by m_blue, we get

1 = -m_red * m_blue

Taking the opposite of both sides tells us that

-1 = m_red * m_blue

The product of the slopes of our lines is -1. In fact, the product of the slopes of any pair of perpendicular lines is -1. So if we have any two perpendicular lines, we can call their slopes m₁ and m₂ and write this fact as an equation:

We can use this equation to check whether or not two lines are perpendicular without trying to measure the angle between them. (Unfortunately, this won't work for vertical lines, because they don't have slopes. Just remember that vertical lines are perpendicular to lines with slopes of 0.)