Perpendicular Lines - SSAT Upper Level Quantitative

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Question

What is the equation of a line that runs perpendicular to the line 2_x_ + y = 5 and passes through the point (2,7)?

Answer

First, put the equation of the line given into slope-intercept form by solving for y. You get y = -2_x_ +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2_x_ + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½_x_ + 6. Rearranged, it is –x/2 + y = 6.

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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

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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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

Find the slope of a line that is perpendicular to the line with the equation $y=\frac{7}{6}x-8$ .

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.

$\frac{7}{6}\rightarrow-\frac{6}{7}$

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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 equations represents a line that goes through the point and is perpendicular to the line ?

Answer

In order to solve this problem, we need first to transform the equation from standard form to slope-intercept form:

Transform the original equation to find its slope.

First, subtract from both sides of the equation.

Simplify and rearrange.

Next, divide both sides of the equation by 6.

$\frac{6y}{6} = -\frac{3x}{6} + \frac{12}{6}$

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

The slope of our first line is equal to $-\frac{1}{2}$ . Perpendicular lines have slopes that are opposite reciprocals of each other; therefore, if the slope of one is x, then the slope of the other is equal to the following:

$-\frac{1}{x}$

Let's calculate the opposite reciprocal of our slope:

$-\frac{1}{2}\rightarrow -\left (-\frac{2}{1} \right )=2$

The slope of our line is equal to 2. We now have the following partial equation:

We are missing the y-intercept, . Substitute the x- and y-values in the given point to solve for the missing y-intercept.

Add 4 to both sides of the equation.

Substitute this value into our partial equation to construct the equation of our line:

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Question

Line m passes through the points (1, 4) and (5, 2). If line p is perpendicular to m, then which of the following could represent the equation for p?

Answer

The slope of m is equal to y2-y1/x2-x1 = 2-4/5-1 = -1/2

Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be –1:

(slope of p) * (-1/2) = -1

Slope of p = 2

So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x – y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.

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Question

What line is perpendicular to x + 3_y_ = 6 and travels through point (1,5)?

Answer

Convert the equation to slope intercept form to get y = –1/3_x_ + 2. The old slope is –1/3 and the new slope is 3. Perpendicular slopes must be opposite reciprocals of each other: _m_1 * _m_2 = –1

With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2

So y = 3_x_ + 2

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Question

Which line below is perpendicular to ?

Answer

The definition of a perpendicular line is one that has a negative, reciprocal slope to another.

For this particular problem, we must first manipulate our initial equation into a more easily recognizable and useful form: slope-intercept form or .

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

According to our formula, our slope for the original line is $-\frac{5}{6}$ . We are looking for an answer that has a perpendicular slope, or an opposite reciprocal. The opposite reciprocal of $-\frac{5}{6}$ is $\frac{6}{5}$ . Flip the original and multiply it by .

Our answer will have a slope of $\frac{6}{5}$ . Search the answer choices for $\frac{6}{5}$ in the position of the equation.

$\dpi{100} y = \frac{6}{5}x + 3$ is our answer.

(As an aside, the negative reciprocal of 4 is $-\frac{1}{4}$ . Place the whole number over one and then flip/negate. This does not apply to the above problem, but should be understood to tackle certain permutations of this problem type where the original slope is an integer.)

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Question

If a line has an equation of $2y=3x+3$ , what is the slope of a line that is perpendicular to the line?

Answer

Putting the first equation in slope-intercept form yields $y=\frac{3}{2}x+\frac{3}{2}$ .

A perpendicular line has a slope that is the negative inverse. In this case, $-\frac{2}{3}$ .

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