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SSAT Upper Level Math : Perpendicular Lines

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Example Questions

Example Question #1 : Perpendicular Lines

Two perpendicular lines intersect at the point (5,4) . One line passes through point (3,1) ; the other passes through point $\left ( 4, Y\right )$ . Evaluate .

Possible Answers:

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

$Y = 4\frac{2}{3}$

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

$Y = 5\frac{1}{2}$

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

Correct answer:

$Y = 4\frac{2}{3}$

Explanation:

The line that passes through $\left ( 3,1\right )$ and $\left ( 5,4\right )$ has slope

$m = \frac{4-1}{5-3} = \frac{3}{2}$ .

The line that passes through $\left ( 4, Y\right )$ and (5,4) , being perpendicular to the first, has as its slope the opposite reciprocal of $\frac{3}{2}$ , or $-\frac{2}{3}$ .

Therefore, to find , we use the slope formula and solve for :

$-\frac{2}{3} = \frac{Y-4}{4-5}$

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

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

$\frac{2}{3}= Y-4$

$\frac{2}{3}+ 4= Y-4 + 4$

$4\frac{2}{3}= Y$

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Example Question #1 : Perpendicular Lines

A line has the following equation:

3y+12x=9

Which of the following could be a line that is perpendicular to this given line?

Possible Answers:

9y-2x=6

3x+8y=12

4y-x=8

3y+9x=20

Correct answer:

4y-x=8

Explanation:

First, put the equation of the given line in the y=mx+b form to find its slope.

3y=-12x+9

y=-4y+3

Since the slope of the given line is , the slope of the line that is perpendicular must be its negative reciprocal, $\frac{1}{4}$ .

Now, put each answer choice in y=mx+b form to see which one has a slope of $\frac{1}{4}$ .

4y-x=8

4y=x+8

$y=\frac{1}{4}x+2$

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Example Question #3 : How To Find Out If Lines Are Perpendicular

Which of the following lines is perpendicular to the line y+3x=7 ?

Possible Answers:

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

12y-4x=-6

6y+2x=14

3y+9x=2

2y-6x=4

Correct answer:

12y-4x=-6

Explanation:

Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.

y+3x=7

y=-3x+7

The slope of this line is . The negative reciprocal will be $+\frac{1}{3}$ , which will be the slope of the perpendicular line.

Now we need to find the answer choice with this slope by converting to slope-intercept form.

12y-4x=-6

12y=4x-6

$y=\frac{4}{12}x-\frac{6}{12}$

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

This equation has a slope of $+\frac{1}{3}$ , and must be our answer.

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

Which of the following lines is perpindicular to y-3x=7

Possible Answers:

$y-\frac{1}{3}x=12$

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

None of the other answers

$y=\frac{-1}{3}x - 4$

y=-3x-2

Correct answer:

$y=\frac{-1}{3}x - 4$

Explanation:

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, y=mx+b , where m is the slope of the line. Two lines are perpindicular to each other if one slope is the negative and reciprocal of the other.

The first step of this problem is to get it into the form, y=mx+b , which is y=3x+7 . Now we know that the slope, m, is . The reciprocal of that is $\frac{1}{3}$ , and the negative of that is $\frac{-1}{3}$ . Therefore, any line that has a slope of $\frac{-1}{3}$ will be perpindicular to the original line.

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

Which of the following equations represents a line that is perpendicular to the line with points (5,-13) and (7,11) ?

Possible Answers:

y=12x-4

$y=\frac{1}{12}x+1$

$y=-\frac{1}{12}x+4$

y=-12x+6

y=-x+4

Correct answer:

$y=-\frac{1}{12}x+4$

Explanation:

If lines are perpendicular, then their slopes will be negative reciprocals.

First, we need to find the slope of the given line.

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{11-(-13)}{7-5}$

$m=\frac{24}{2}$

m=12

Because we know that our given line's slope is , the slope of the line perpendicular to it must be $-\frac{1}{12}$ .

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Example Question #3 : Perpendicular Lines

Which of the following lines is perpendicular to a line with a slope m=7 ?

Possible Answers:

$y=\frac{1}{7}x+9$

$y=-\frac{1}{7}x+9$

Not enough information provided

y=-7x+9

y=7x+9

Correct answer:

$y=-\frac{1}{7}x+9$

Explanation:

For a given line y=mx+b with a slope , any perpendicular line would have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Given that m=7 in this instance, we can conclude that the slope of a perpendicular line would be $-\frac{1}{m}=-\frac{1}{7}$ . Therefore, the equation that contains this slope is $y=-\frac{1}{7}x+9$ .

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Example Question #4 : Perpendicular Lines

Which of the following lines would be perpendicular to $y=\frac{5}{4}x+12$ ?

Possible Answers:

$y=-\frac{5}{4}x+12$

$y=-\frac{4}{5}x+12$

Not enough information provided to solve

$y=\frac{4}{5}x+12$

$y=12x+\frac{4}{5}$

Correct answer:

$y=-\frac{4}{5}x+12$

Explanation:

For a given line y=mx+b with a slope , any perpendicular line would have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Given that $m=\frac{5}{4}$ in this instance, we can conclude that the slope of a perpendicular line would be $-\frac{1}{m}=-\frac{1}{\frac{5}{4}}=-\frac{4}{5}$ . Given the perpendicular slope, we can now conclude that the perpendicular line is $y=-\frac{4}{5}x+12$ .

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Example Question #3 : Perpendicular Lines

A given line has the equation $y=\frac{3}{4}x+7$ . What is the slope of any line that is perpendicular to this line?

Possible Answers:

$-\frac{3}{4}$

$-\frac{4}{3}$

$-\frac{1}{7}$

Correct answer:

$-\frac{4}{3}$

Explanation:

For a given line y=mx+b with a slope , any perpendicular line would have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Given that $m=\frac{3}{4}$ in this instance, we can conclude that the slope of a perpendicular line would be $-\frac{1}{m}=-\frac{1}{\frac{3}{4}}=-\frac{4}{3}$ .

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Example Question #2 : Perpendicular Lines

The equation for one line is y=-2x+1 . What is the slope of the line that is perpendicular to this line?

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

A line is perpendicular to another if their slopes are negative reciprocals of each other.

Since the slope of the given line is $-\frac{2}{1}$ , the negative reciprocal would be $\frac{1}{2}$ .

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Example Question #1 : How To Find The Slope Of A Perpendicular Line

What is the slope of any line perpendicular to 2y = 4x +3 ?

Possible Answers:

– 4

– ½

Correct answer:

– ½

Explanation:

First, we must solve the equation for y to determine the slope: y = 2x + ³/₂

By looking at the coefficient in front of x, we know that the slope of this line has a value of 2. To fine the slope of any line perpendicular to this one, we take the negative reciprocal of it:

slope = m , perpendicular slope = – ¹/_m

slope = 2 , perpendicular slope = – ¹/₂

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SSAT Upper Level Math : Perpendicular Lines

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #1 : Perpendicular Lines

Example Question #1 : Perpendicular Lines

Example Question #3 : How To Find Out If Lines Are Perpendicular

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

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

Example Question #3 : Perpendicular Lines

Example Question #4 : Perpendicular Lines

Example Question #3 : Perpendicular Lines

Example Question #2 : Perpendicular Lines

Example Question #1 : How To Find The Slope Of A Perpendicular Line

All SSAT Upper Level Math Resources

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