How to find out if lines are perpendicular - SSAT Upper Level Quantitative

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Question

Which of the following lines is perpendicular to the line ?

Answer

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.

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.

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

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

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

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Question

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

Answer

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 , 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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Question

Which of the following equations represents a line that is perpendicular to the line with points and ?

Answer

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

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

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

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

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

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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Question

Which of the following lines is perpindicular to

Answer

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, , 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, , which is . 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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Question

A line has the following equation:

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

Answer

First, put the equation of the given line in the form to find its slope.

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 form to see which one has a slope of $\frac{1}{4}$ .

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

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Question

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?

Answer

For a given line 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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Question

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

Answer

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

Given that 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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Question

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

Answer

For a given line 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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Question

Which of the following lines is perpendicular to the line ?

Answer

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.

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.

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

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

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

Compare your answer with the correct one above

Question

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

Answer

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 , 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 :

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

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

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

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

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

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

Compare your answer with the correct one above

Question

Which of the following equations represents a line that is perpendicular to the line with points and ?

Answer

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

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

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

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

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

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

Compare your answer with the correct one above

Question

Which of the following lines is perpindicular to

Answer

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, , 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, , which is . 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.

Compare your answer with the correct one above

Question

A line has the following equation:

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

Answer

First, put the equation of the given line in the form to find its slope.

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 form to see which one has a slope of $\frac{1}{4}$ .

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

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Question

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?

Answer

For a given line 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 $-\frac{1}{m}=-\frac{1}{\frac{3}{4}}=-\frac{4}{3}$ .

Compare your answer with the correct one above

Question

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

Answer

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

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

Compare your answer with the correct one above

Question

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

Answer

For a given line 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 $y=-\frac{4}{5}x+12$ .

Compare your answer with the correct one above

Question

Which of the following lines is perpendicular to the line ?

Answer

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.

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.

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

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

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

Compare your answer with the correct one above

Question

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

Answer

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 , 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 :

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

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

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

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

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

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

Compare your answer with the correct one above

Question

Which of the following equations represents a line that is perpendicular to the line with points and ?

Answer

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

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

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

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

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

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

Compare your answer with the correct one above

Question

Which of the following lines is perpindicular to

Answer

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, , 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, , which is . 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.

Compare your answer with the correct one above

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