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Algebra 1 : Perpendicular Lines

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Example Questions

Example Question #1 : Perpendicular Lines

Which ONE of these statements about the lines defined by the following equations is TRUE?

Line 1: $y=\frac{2}{3}x+ 12$

Line 2: $y= -\frac{3}{2}x - 12$

Possible Answers:

The slopes of the two lines are identical.

The lines intersect only once because they are parallel.

The lines intersect at the point (0,-8) .

The lines do not intersect.

The lines intersect and are perpendicular.

Correct answer:

The lines intersect and are perpendicular.

Explanation:

The TRUE statement:

"The lines intersect and are perpendicular." This is true because the slopes of the two lines are opposite-reciprocals of each other.

$m_1=\frac{2}{3}\ \textup{and}\ m_2=-\frac{3}{2}$

The FALSE statements:

"The lines intersect at the point (0,-8) ." The lines actually intersect at the point (-8,0) . Neither line touches the point (0,-8) , as their y-intercepts are given in their respective equations as (0,12) and (0,-12) .

"The slopes of the two lines are identical." This is not true because the slope of Line 1 is $\frac{2}{3}$ whereas the slope of Line 2 is $-\frac{3}{2}$ .

"The lines do not intersect." The lines would need to be parallel (i.e., have the same slope) for this to be the case, but the lines do not have the same slope.

"The lines intersect only once because they are parallel." Parallel lines never intersect, so this statement cannot be made of any set of two lines.

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Example Question #1 : Perpendicular Lines

Determine if the lines y=x+3 and y=-x+10 are perpendicular.

Possible Answers:

The lines are perpendicular

There is not enough information to determine the answer

The lines are not perpendicular

Correct answer:

The lines are perpendicular

Explanation:

For lines to be perpendicular, the slopes need to be negative reciprocals of each other. For the line y=x+3 , the slope is 1. For a line to be perpendicular to it, it will need to have a slope of $\frac{-1}{1}=1$ . Since the line y=-x+10 has a slope of -1, the lines are perpendicular to each other.

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Example Question #3 : Perpendicular Lines

Which of these lines is perpendicular to

$4y-2x=12\ ?$

Possible Answers:

$y=-\frac{1}{2}x+6$

None of the other answers

$y=\frac{1}{2}x-8$

y=-2x+9

y=2x+3

Correct answer:

y=-2x+9

Explanation:

Perpendicular lines have slopes that are negative reciprocals of each other. If you convert the given line to the y=mx+b form, you get

$y=\frac{1}{2}x+3$

which indicates a slope of $\frac{1}{2}$ . Thus, the slope of the perpendicular line must be , which is the negative reciprocal of $\frac{1}{2}$ . The only line with a slope of is

y=-2x+9 .

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Example Question #3 : Equations Of Lines

Which of the following equations describes a line perpendicular to the line x=3 ?

Possible Answers:

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

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

y=2

y=3x

$x=\frac{1}{3}$

Correct answer:

y=2

Explanation:

The line x=3 is a vertical line. Therefore, a perpendicular line is going to be horizontal and have a slope of zero.

The equation y=2 is such a line.

The lines $x=\frac{1}{3}$ and $x=-\frac{1}{3}$ are both vertical lines, while the lines y=3x and $y=-\frac{x}{3}$ have slopes of and $-\frac{1}{3}$ , respectively.

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Example Question #1 : Perpendicular Lines

Which of these lines is perpendicular to y=-3x+6 ?

Possible Answers:

y=3x-2

y=-3x+8

y=x+5

$y=-\frac{1}{3}x-7$

$y=\frac{1}{3}x-9$

Correct answer:

$y=\frac{1}{3}x-9$

Explanation:

Perpendicular lines have slopes that are negative reciprocals of one another. Since all of these lines are in the y=mx+b format, it is easy to determine their slopes, or .

The slope of the original line is , so any line that is perpendicular to it must have a slope of $\frac{1}{3}$ .

The only line with a slope of $\frac{1}{3}$ is $y=\frac{1}{3}x-9$ .

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Example Question #2 : Perpendicular Lines

Which of the following lines could be perpendicular to the following:

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

Possible Answers:

$y=-1\frac{1}{3}x-128$

None of the available answers

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

y=0

$y=\frac{4}{6}x-17$

Correct answer:

$y=-1\frac{1}{3}x-128$

Explanation:

The only marker for whether lines are perpendicular is whether their slopes are the opposite-reciprocal for the other line's slope. The -intercept is not important. Therefore, the line perpendicular to $y=\frac{3}{4}x-2$ will have a slope of $m=-\frac{4}{3}$ or $m=-1\frac{1}{3}$

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Example Question #5 : Perpendicular Lines

Find a line perpendicular to the line with the equation:

y=-8x+1

Possible Answers:

y=-8x-1

$y=\frac{1}{8}x-19$

$y=-\frac{1}{8}x-1$

x=8x+2

Correct answer:

$y=\frac{1}{8}x-19$

Explanation:

Lines can be written in the slope-intercept format:

y=mx+b

In this format, equals the line's slope and represents where the line intercepts the y-axis.

In the given equation:

m=-8

Perpendicular lines have slopes that are negative reciprocals of each other.

First, we need to find its reciprocal.

$-8\rightarrow-\frac{1}{8}$

Second, we need to rewrite it with the opposite sign.

$-\frac{1}{8}\rightarrow\frac{1}{8}$

Only one of the choices has a slope of $\frac{1}{8}$ .

$y=\frac{1}{8}x-19$

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Example Question #6 : Perpendicular Lines

Find a line perpendicular to the line with the equation:

$y=\frac{5}{6}x-1$

Possible Answers:

$y=-\frac{6}{5}x+9$

$y=-\frac{5}{6}x$

$y=\frac{5}{6}x-8$

$y=\frac{6}{5}x-78$

Correct answer:

$y=-\frac{6}{5}x+9$

Explanation:

Lines can be written in the slope-intercept format:

y=mx+b

In this format, equals the line's slope and represents where the line intercepts the y-axis.

In the given equation:

$m=\frac{5}{6}$

Perpendicular lines have slopes that are negative reciprocals of each other.

First, we need to find its reciprocal.

$\frac{5}{6}\rightarrow\frac{6}{5}$

Second, we need to rewrite it with the opposite sign.

$\frac{6}{5}\rightarrow-\frac{6}{5}$

Only one of the choices has a slope of $-\frac{6}{5}$ .

$y=-\frac{6}{5}x+9$

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Example Question #1 : How To Find Out If Lines Are Perpendicular

Find a line perpendicular to the line with the equation:

$y=-\frac{1}{15}x-12$

Possible Answers:

y=-15x+6

$y=\frac{1}{15}x+6$

$y=-\frac{1}{15}x+6$

y=15x+6

Correct answer:

y=15x+6

Explanation:

Lines can be written in the slope-intercept format:

y=mx+b

In this format, equals the line's slope and represents where the line intercepts the y-axis.

In the given equation:

$m=-\frac{1}{15}$

Perpendicular lines have slopes that are negative reciprocals of each other.

First, we need to find its reciprocal.

$-\frac{1}{15}\rightarrow-\frac{15}{1}$

Rewrite.

$-\frac{15}{1}=-15$

Second, we need to rewrite it with the opposite sign.

$-15\rightarrow15$

Only one of the choices has a slope of .

y=15x+6

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Example Question #10 : Perpendicular Lines

Find a line perpendicular to the line with the equation:

$y=\frac{1}{2}x+79$

Possible Answers:

$y=\frac{1}{2}x-10$

y=-2x-10

y=2x-10

$y=-\frac{1}{2}x-10$

Correct answer:

y=-2x-10

Explanation:

Lines can be written in the slope-intercept format:

y=mx+b

In this format, equals the line's slope and represents where the line intercepts the y-axis.

In the given equation:

$m=\frac{1}{2}$

Perpendicular lines have slopes that are negative reciprocals of each other.

First, we need to find its reciprocal.

$\frac{1}{2}\rightarrow\frac{2}{1}$

Rewrite.

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

Second, we need to rewrite it with the opposite sign.

$2\rightarrow-2$

Only one of the choices has a slope of .

y=-2x-10

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Algebra 1 : Perpendicular Lines

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : Perpendicular Lines

Example Question #1 : Perpendicular Lines

Example Question #3 : Perpendicular Lines

Example Question #3 : Equations Of Lines

Example Question #1 : Perpendicular Lines

Example Question #2 : Perpendicular Lines

Example Question #5 : Perpendicular Lines

Example Question #6 : Perpendicular Lines

Example Question #1 : How To Find Out If Lines Are Perpendicular

Example Question #10 : Perpendicular Lines

All Algebra 1 Resources

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