1.5 Parallel and Perpendicular Lines

Parallel Lines

Two lines in an xy-plane are said to be parallel lines if they do not cross each other. We can identify whether two lines are parallel or not by looking at their slopes. Two distinct nonvertical lines are parallel if and only if their slopes are equal.

Suppose there are two lines. One line goes through the two points (-1, 1) and (2, 7). The other line passes through the two points (0, 4) and (-2, 0). Are the two lines are parallel? To find out, let’s calculate the slope of each line.

The slope of the first line is

$m = \frac{1 - 7}{- 1 - 2} = \frac{- 6}{- 3} = 2$

The slope of the second line is

$m = \frac{4 - 0}{0 - (- 2)} = \frac{4}{2} = 2$

Since both lines have the same slope, they are parallel.

Figure 1.7: Parallel Lines

Examples

Let’s practice identifying parallel lines with a few examples.

Example 1

Suppose there are two lines. One line passes through the points (3, 5) and (-1, 8), and the other passes through (1, 4) and (3, 1). Are these two lines parallel?

In order to determine whether they are parallel, we need to find the slope of each line.

The slope of the first line is

$m = \frac{5 - 8}{3 - (- 1)} = \frac{- 3}{4} = - \frac{3}{4}$

The slope of the second line is

$m = \frac{4 - 1}{1 - 3} = \frac{3}{- 2} = - \frac{3}{2}$

Since the slopes are not the same, the lines are not parallel.

Example 2

Suppose there are two lines. One line passes through the points (-1, 0) and (5, 12), and the other passes through (1, 2) and (-2, -4). Are these two lines parallel?

In order to determine whether they are parallel, we need to find the slope of each line.

The slope of the first line is

$m = \frac{0 - 12}{- 1 - 5} = \frac{- 12}{- 6} = 2$

The slope of the second line is

$m = \frac{2 - (- 4)}{1 - (- 2)} = \frac{6}{3} = 2$

Since the slopes are the same, the lines are parallel.

Perpendicular Lines

Two lines are perpendicular lines if they intersect at a right angle. Similarly to identifying whether lines are parallel, we can identify whether two lines are perpendicular based on the slopes of the lines. Two distinct nonvertical lines are perpendicular if and only if the product of their slopes is -1. In other words, two lines are perpendicular if one line’s slope is the opposite of the reciprocal of the other. For example, lines with slopes of $\frac{1}{3}$ and -3 are perpendicular. That is because -3 is the opposite of the reciprocal of $\frac{1}{3}$ . Mathematically, suppose that slopes of two lines are $m_{1}$ and $m_{2}$ . The lines are perpendicular if and only if

$m_{1} m_{2} = - 1$

This means that if you are given the function of a line, $y = m_{1} x + b$ , and you are looking for the function of a line that is perpendicular to it, $y = m_{2} x + b$ , the slope $m_{2}$ is

$m_{2} = - \frac{1}{m_{1}}$

Suppose there are two lines. One line goes through the two points (-1, 1) and (2, 7). The other line passes through the two points (0, 4) and (-4, 6). How can we determine whether the two lines are perpendicular? The answer is to find the slope of each line, and then multiply them together.

The slope of the first line is

$m = \frac{1 - 7}{- 1 - 2} = \frac{- 6}{- 3} = 2$

The slope of the second line is

$m = \frac{4 - 6}{0 - (- 4)} = \frac{- 2}{4} = - \frac{1}{2}$

Next, we find the product of the two slopes by multiplying them together:

$2 (- \frac{1}{2}) = - 1$

Since the product of the two slopes is -1, the two lines are perpendicular.

Figure 1.8: Perpendicular Lines

Examples

Let’s practice identifying perpendicular lines with a few examples.

Example 1

Are these two lines perpendicular?

$\begin{aligned} 3 x - 2 y = 4 \\ 2 x + 3 y = 11 \end{aligned}$

To figure out whether the lines are perpendicular, we need to find the slope of each line. First, we will rewrite these general-form equations in slope-intercept form.

$3 x - 2 y = 4$
$2 y = 3 x - 4$
$y = \frac{3}{2} x - 2$

$2 x + 3 y = 11$
$3 y = - 2 x + 11$
$y = - \frac{2}{3} + \frac{11}{3}$

Next, we find the product of the two slopes:

$\frac{3}{2} (- \frac{2}{3}) = - 1$

Since the product of the slopes is -1, the lines are perpendicular.

Example 2

Suppose there are two lines. The first line passes through the points (1, 4) and (5, 1), and the second line passes through the points (-2, -4) and (2, 4). Are the lines perpendicular?

Since two points for each line are given, we will first find the slope of each line.

$m_{1} = \frac{1 - 5}{4 - 1} = \frac{- 4}{3} = - \frac{4}{3}$

$m_{2} = \frac{- 2 - 2}{- 4 - 4} = \frac{- 4}{- 8} = \frac{1}{2}$

Next, we find the product of the slopes:

$(- \frac{4}{3}) \frac{1}{2} = - \frac{2}{3}$

Since the product of the slopes is not -1, the lines are not perpendicular.

Example 3

Find the equation of a line that is perpendicular to the line $y = 3 x - 2$ and that passes through the point (2, 1).

First, we need to find the slope $m$ of the line that is perpendicular to $y = 3 x - 2$ .

$m = - \frac{1}{3}$

Since we know the line goes through (2, 1), we can plug this point into the point-slope equation:

$y - 1 = - \frac{1}{3} (x - 2)$

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