Perpendicular Lines - 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

What is the slope of any line perpendicular to 2_y_ = 4_x_ +3 ?

Answer

First, we must solve the equation for y to determine the slope: y = 2_x_ + 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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Question

What line is perpendicular to 2x + y = 3 at (1,1)?

Answer

Find the slope of the given line. The perpendicular slope will be the opposite reciprocal of the original slope. Use the slope-intercept form (y = mx + b) and substitute in the given point and the new slope to find the intercept, b. Convert back to standard form of an equation: ax + by = c.

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Question

What is the slope of the line perpendicular to the line given by the equation

6x – 9y +14 = 0

Answer

First rearrange the equation so that it is in slope-intercept form, resulting in y=2/3 x + 14/9. The slope of this line is 2/3, so the slope of the line perpendicular will have the opposite reciprocal as a slope, which is -3/2.

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Question

What is the slope of the line perpendicular to the line represented by the equation y = -2x+3?

Answer

Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2.

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Question

What is the slope of a line perpendicular to the following:

$y-3=\frac{1}{2}(x-4)$

Answer

The question puts the line in point-slope form y – y1 = m(x – x1), where m is the slope. Therefore, the slope of the original line is 1/2. A line perpendicular to another has a slope that is the negative reciprocal of the slope of the other line. The negative reciprocal of the original line is _–_2, and is thus the slope of its perpendicular line.

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Question

A line is defined by the following equation:

What is the slope of a line that is perpendicular to the line above?

Answer

The equation of a line is where is the slope.

Rearrange the equation to match this:

$y=\frac{2}{4}x-\frac{6}{4}$

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

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

For the perpendicular line, the slope is the negative reciprocal;

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

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Question

Find the slope of a line perpendicular to the line y = –3x – 4.

Answer

First we must find the slope of the given line. The slope of y = –3x – 4 is –3. The slope of the perpendicular line is the negative reciprocal. This means you change the sign of the slope to its opposite: in this case to 3. Then find the reciprocal by switching the denominator and numerator to get 1/3; therefore the slope of the perpendicular line is 1/3.

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Question

The equation for one line is . What is the slope of the line that is perpendicular to this line?

Answer

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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Question

Find the slope of the line perpendicular to the line that has the equation .

Answer

Perpendicular lines have slopes that are negative reciprocals of each other. In other words, change the sign and flip the fraction around.

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

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Question

Find the slope of a line that is perpendicular to the line with the equation $y=\frac{1}{5}x-2$ .

Answer

Perpendicular lines have slopes that are negative reciprocals of each other. In other words, change the sign and flip the fraction around to find the slope of the line perpendicular to the given one.

$\frac{1}{5}\rightarrow-5$

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Question

Find the slope of a line perpendicular to the line with the equation .

Answer

Perpendicular lines have slopes that are negative reciprocals of each other. In other words, change the sign and flip the fraction around to find the slope of the line perpendicular to the given one.

$-9\rightarrow\frac{1}{9}$

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Question

Find the slope of a line that is perpendicular to the line with the equation $y=\frac{3}{2}x-1$ .

Answer

Perpendicular lines have slopes that are negative reciprocals of each other. In other words, change the sign and flip the fraction around to find the slope of the line perpendicular to the given one.

$\frac{3}{2}\rightarrow-\frac{2}{3}$

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