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# 3.9: Perpendicular Lines in the Coordinate Plane

Difficulty Level: At Grade Created by: CK-12
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Practice Perpendicular Lines in the Coordinate Plane
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What if you wanted to figure out if two lines in a shape met at right angles? How could you do this? After completing this Concept, you'll be able to use slope to help you determine whether or not lines are perpendicular.

### Watch This

Watch the portion of this video that deals with Perpendicular Lines

### Guidance

Recall that the definition of perpendicular is two lines that intersect at a $90^\circ$ , or right, angle. In the coordinate plane, that would look like this:

If we take a closer look at these two lines, we see that the slope of one is -4 and the other is $\frac{1}{4}$ . This can be generalized to any pair of perpendicular lines in the coordinate plane. The slopes of perpendicular lines are opposite signs and reciprocals of each other.

#### Example A

Find the slope of the perpendicular lines to the lines below.

a) $y=2x+3$

b) $y=-\frac{2}{3}x-5$

c) $y=x+2$

We are only concerned with the slope for each of these.

a) $m = 2$ , so $m_\perp$ is the reciprocal and negative, $m_\perp=-\frac{1}{2}$ .

b) $m=-\frac{2}{3}$ , take the reciprocal and make the slope positive, $m_\perp=\frac{3}{2}$ .

c) Because there is no number in front of $x$ , the slope is 1. The reciprocal of 1 is 1, so the only thing to do is make it negative, $m_\perp=-1$ .

#### Example B

Find the equation of the line that is perpendicular to $y=-\frac{1}{3}x+4$ and passes through (9, -5).

First, the slope is the reciprocal and opposite sign of $-\frac{1}{3}$ . So, $m = 3$ . Now, we need to find the $y-$ intercept. 4 is the $y-$ intercept of the given line, not our new line . We need to plug in 9 for $x$ and -5 for $y$ to solve for the new $y-$ intercept $(b)$ .

$-5 & = 3(9)+b\\-5 & = 27 + b \qquad \text{Therefore, the equation of line is} \ y=3x-32.\\-32 & = b$

#### Example C

Graph $3x-4y=8$ and $4x+3y=15$ . Determine if they are perpendicular.

First, we have to change each equation into slope-intercept form. In other words, we need to solve each equation for $y$ .

$3x-4y & = 8 && 4x+3y = 15\\-4y & = -3x+8 && 3y = -4x + 15\\y & = \frac{3}{4}x-2 && y = -\frac{4}{3}x+5$

Now that the lines are in slope-intercept form (also called $y-$ intercept form), we can tell they are perpendicular because their slopes are opposite reciprocals.

Watch this video for help with the Examples above.

### Vocabulary

Two lines in the coordinate plane with slopes that are opposite signs and reciprocals of each other are perpendicular and intersect at a $90^\circ$ , or right, angle. Slope measures the steepness of a line.

### Guided Practice

1. Determine which of the following pairs of lines are perpendicular.

• $y=-2x+3$ and $y=\frac{1}{2}x+3$
• $y=4x-2$ and $y=4x+5$
• $y=-x+5$ and $y=x+1$

2. Find the equation of the line that is perpendicular to the line $y=2x+7$ and goes through the point (2, -2).

3. Give an example of a line that is perpendicular to the line $y=\frac{2}{3}x-4$ .

1. Two lines are perpendicular if their slopes are opposite reciprocals. The only pairs of lines this is true for is the first pair, because $-2$ and $\frac{1}{2}$ are opposites and reciprocals.

2. The perpendicular line goes through (2, -2), but the slope is $-\frac{1}{2}$ because we need to take the opposite reciprocal of $2$ .

$y & = -\frac{1}{2}x+b\\-2 & = -\frac{1}{2}(2) + b\\-2 & = -1+b\\-1 & =b$

The equation is $y = -\frac{1}{2}x-1$ .

3. Any line perpendicular to $y=\frac{2}{3}x-4$ will have a slope of $-\frac{3}{2}$ . Any equation of the form $y=-\frac{3}{2}x+b$ will work.

### Practice

1. Determine which of the following pairs of lines are perpendicular.
1. $y=-3x+1$ and $y=3x-1$
2. $2x-3y=6$ and $3x+2y=6$
3. $5x+2y=-4$ and $5x+2y=8$
4. $x-3y=-3$ and $x+3y=9$
5. $x+y=6$ and $4x+4y=-16$

Determine the equation of the line that is perpendicular to the given line, through the given point.

1. $y=x-1; \ (-6, \ 2)$
2. $y=3x+4; \ (9, \ -7)$
3. $5x-2y=6; \ (5, \ 5)$
4. $y = 4; \ (-1, \ 3)$
5. $x = -3; \ (1, \ 8)$
6. $x - 3y = 11; \ (0, \ 13)$

Determine if each pair of lines is perpendicular or not.

For the line and point below, find a perpendicular line, through the given point.

Jul 17, 2012

Feb 18, 2015