### Comparing Equations of Parallel and Perpendicular Lines

In this section you will learn how **parallel lines** and **perpendicular lines** are related to each other on the coordinate plane. Let’s start by looking at a graph of two parallel lines.

We can clearly see that the two lines have different \begin{align*}y-\end{align*}intercepts: 6 and –4.

How about the slopes of the lines? The slope of line \begin{align*}A\end{align*} is \begin{align*}\frac{6-2}{0-(-2)}=\frac{4}{2}=2\end{align*}, and the slope of line \begin{align*}B\end{align*} is \begin{align*}\frac{0-(-4)}{2-0}=\frac{4}{2}=2\end{align*}. The slopes are the same.

Is that significant? Yes. By definition, parallel lines never meet. That means that when one of them slopes up by a certain amount, the other one has to slope up by the same amount so the lines will stay the same distance apart. If you look at the graph above, you can see that for any \begin{align*}x-\end{align*}value you pick, the \begin{align*}y-\end{align*}values of lines \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are the same vertical distance apart—which means that both lines go up by the same vertical distance every time they go across by the same horizontal distance. In order to stay parallel, their slopes must stay the same.

**All parallel lines** have the same slopes and different \begin{align*}y-\end{align*}intercepts.

Now let’s look at a graph of two perpendicular lines.

We can’t really say anything about the \begin{align*}y-\end{align*}intercepts. In this example, the \begin{align*}y-\end{align*}intercepts are different, but if we moved the lines four units to the right, they would both intercept the \begin{align*}y-\end{align*}axis at (0, -2). So perpendicular lines can have the same or different \begin{align*}y-\end{align*}intercepts.

What about the relationship between the slopes of the two lines?

To find the slope of line \begin{align*}A\end{align*}, we pick two points on the line and draw the blue (upper) right triangle. The legs of the triangle represent the rise and the run. We can see that the slope is \begin{align*}\frac{8}{4}\end{align*}, or 2.

To find the slope of line \begin{align*}B\end{align*}, we pick two points on the line and draw the red (lower) right triangle. Notice that the two triangles are identical, only rotated by \begin{align*}90^\circ\end{align*}. Where line \begin{align*}A\end{align*} goes 8 units up and 4 units right, line \begin{align*}B\end{align*} goes 8 units right and 4 units down. Its slope is \begin{align*}-\frac{4}{8}\end{align*}, or \begin{align*}-\frac{1}{2}\end{align*}.

This is always true for perpendicular lines; where one line goes \begin{align*}a\end{align*} units up and \begin{align*}b\end{align*} units right, the other line will go \begin{align*}a\end{align*} units right and \begin{align*}b\end{align*} units down, so the slope of one line will be \begin{align*}\frac{a}{b}\end{align*} and the slope of the other line will be \begin{align*}-\frac{b}{a}\end{align*}.

The slopes of **perpendicular lines** are always negative reciprocals of each other.

**Determining When Lines are Parallel or Perpendicular**

You can find whether lines are parallel or perpendicular by comparing the slopes of the lines. If you are given points on the lines, you can find their slopes using the formula. If you are given the equations of the lines, re-write each equation in a form that makes it easy to read the slope, such as the slope-intercept form.

1. Determine whether the lines are parallel or perpendicular or neither. One line passes through the points (2, 11) and (-1, 2); another line passes through the points (0, -4) and (-2, -10).

Find the slope of each line and compare them.

\begin{align*}m_1=\frac{2-11}{-1-2}=\frac{-9}{-3}=3 \ \quad \ \text{and} \ \quad \ m_2=\frac{-10-(-4)}{-2-0}=\frac{-6}{-2}=3\end{align*}

The slopes are equal, so **the lines are parallel.**

2. Determine whether the lines are parallel or perpendicular or neither. One line passes through the points (-2, -7) and (1, 5); another line passes through the points (4, 1) and (-8, 4).

\begin{align*}m_1=\frac{5-(-7)}{1-(-2)}=\frac{12}{3}=4 \quad \ \text{and} \ \quad m_2=\frac{4-1}{-8-4}=\frac{3}{-12}=-\frac{1}{4}\end{align*}

The slopes are negative reciprocals of each other, so **the lines are perpendicular.**

3. Determine whether the lines are parallel or perpendicular or neither. One line pass*es through the points (3, 1) and (-2, -2); another line passes through the points (5, 5) and (4, -6).*

\begin{align*}m_1=\frac{-2-1}{-2-3}=\frac{-3}{-5}=\frac{3}{5} \ \quad \ \text{and} \ \quad \ m_2=\frac{-6-5}{4-5}=\frac{-13}{-1}=13\end{align*}

The slopes are not the same or negative reciprocals of each other, so **the lines are neither parallel nor perpendicular.**

### Examples

Determine whether the lines are parallel or perpendicular or neither:

#### Example 1

\begin{align*}3x+4y=2\end{align*} and \begin{align*}8x-6y=5\end{align*}

Write each equation in slope-intercept form:

line 1: \begin{align*}3x+4y=2 \Rightarrow 4y=-3x+2 \Rightarrow y=-\frac{3}{4}x+\frac{1}{2} \Rightarrow \ \text{slope} = - \frac{3}{4}\end{align*}

line 2: \begin{align*}8x-6y=5 \Rightarrow 8x-5=6y \Rightarrow y=\frac{8}{6} x-\frac{5}{6} \Rightarrow y=\frac{4}{3} x-\frac{5}{6} \Rightarrow \ \text{slope} = \frac{4}{3}\end{align*}

The slopes are negative reciprocals of each other, so **the lines are perpendicular.**

#### Example 2

\begin{align*}2x=y-10\end{align*} and \begin{align*}y=-2x+5\end{align*}

line 1: \begin{align*}2x=y-10 \Rightarrow y=2x+10 \Rightarrow \ \text{slope} = 2\end{align*}

line 2: \begin{align*}y=-2x+5 \Rightarrow \ \text{slope} = -2\end{align*}

The slopes are not the same or negative reciprocals of each other, so **the lines are neither parallel nor perpendicular.**

#### Example 3

\begin{align*}7y+1=7x\end{align*} and \begin{align*}x+5=y\end{align*}

line 1: \begin{align*}7y+1=7x \Rightarrow 7y=7x-1 \Rightarrow y=x-\frac{1}{7} \Rightarrow \ \text{slope} = 1\end{align*}

line 2: \begin{align*}x+5=y \Rightarrow y=x+5 \Rightarrow \ \text{slope} = 1\end{align*}

The slopes are the same, so **the lines are parallel.**

### Review

For 1-10, determine whether the lines are parallel, perpendicular or neither.

- One line passes through the points (-1, 4) and (2, 6); another line passes through the points (2, -3) and (8, 1).
- One line passes through the points (4, -3) and (-8, 0); another line passes through the points (-1, -1) and (-2, 6).
- One line passes through the points (-3, 14) and (1, -2); another line passes through the points (0, -3) and (-2, 5).
- One line passes through the points (3, 3) and (-6, -3); another line passes through the points (2, -8) and (-6, 4).
- One line passes through the points (2, 8) and (6, 0); another line has the equation \begin{align*}x-2y=5\end{align*}.
- One line passes through the points (-5, 3) and (2, -1); another line has the equation \begin{align*}2x+3y=6\end{align*}.
- Both lines pass through the point (2, 8); one line also passes through (3, 5), and the other line has slope 3.
- Line 1: \begin{align*}4y+x=8\end{align*} Line 2: \begin{align*}12y+3x=1\end{align*}
- Line 1: \begin{align*}5y+3x=1\end{align*} Line 2: \begin{align*}6y+10x=-3\end{align*}
- Line 1: \begin{align*}2y-3x+5=0\end{align*} Line 2: \begin{align*}y+6x=-3\end{align*}
- Lines \begin{align*}A, B, C, D\end{align*}, and \begin{align*}E\end{align*} all pass through the point (3, 6). Line \begin{align*}A\end{align*} also passes through (7, 12); line \begin{align*}B\end{align*} passes through (8, 4); line \begin{align*}C\end{align*} passes through (-1, -3); line \begin{align*}D\end{align*} passes through (1, 1); and line \begin{align*}E\end{align*}passes through (6, 12).
- Are any of these lines perpendicular? If so, which ones? If not, why not?
- Are any of these lines parallel? If so, which ones? If not, why not?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 5.4.