4.5 Use the slope–intercept form of an equation of a line  (Page 7/12)

Use slopes to identify perpendicular lines

Let’s look at the lines whose equations are $y=\frac{1}{4}x-1$ and $y=\mathrm{-4}x+2$ , shown in [link] .

These lines lie in the same plane and intersect in right angles. We call these lines perpendicular .

What do you notice about the slopes of these two lines? As we read from left to right, the line $y=\frac{1}{4}x-1$ rises, so its slope is positive. The line $y=\mathrm{-4}x+2$ drops from left to right, so it has a negative slope. Does it make sense to you that the slopes of two perpendicular lines will have opposite signs?

If we look at the slope of the first line, $m_1=\frac{1}{4}$ , and the slope of the second line, $m_2=\mathrm{-4}$ , we can see that they are negative reciprocals of each other. If we multiply them, their product is $\mathrm{-1}.$

This is always true for perpendicular lines    and leads us to this definition.

We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines.

We find the slope–intercept form of the equation, and then see if the slopes are negative reciprocals. If the product of the slopes is $\mathrm{-1}$ , the lines are perpendicular. Perpendicular lines may have the same y -intercepts.