Perpendicular Lines - ACT Math

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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

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

Which of the following lines is perpendicular to the line with the given equation:
?

Answer

First we must recognize that the equation is given in slope-intercept form, where is the slope of the line.

Two lines are perpendicular if and only if the product of their slopes is . In other words, the slope of a line that is perpendicular to a given line is the negative reciprocal of that slope.

Thus, for a line with a given slope of 3, the line perpendicular to that slope must be the negative reciprocal of 3, or $-\frac{1}{3}$ .

To double check that that does indeed give a product of when multiplied by three simply compute the product:

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

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Question

Are the following two lines parallel, perpendicular, or neither:

and

Answer

Perpendicular lines have slopes whose product is .

The slope is controlled by the coefficient, from the genral form of the slope-intercept equation:

Thus the two lines are perpendicular because:

has

and

has

which when multiplied together results in,

.

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Question

Are the following two lines perpendicular:
$y = -7x + -4\textup{ and }y=\frac{1}{7}x-7\textup{?}$

Answer

For two lines to be perpendicular, their slopes have to have a product of . Find the slopes by the coefficient in front of the .

$-7*\frac{1}{7}= - 1$ and so the two lines are perpendicular. The y-intercept does not matter for determine perpendicularity.

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Question

Are the following two lines perpendicular:

and

Answer

For two lines to be perpendicular they have to have slopes that multiply to get . The slope is found from the in the general equation: .

For the first line, and for the second . and so the lines are not perpendicular.

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Question

Are the lines described by the equations $y=-\frac{2}{3}x +7$ and $y=-\frac{3}{2}x +2$ perpendicular to one another? Why or why not?

Answer

If the slopes of two lines can be calculated, an easy way to determine whether they are perpendicular is to multiply their slopes. If the product of the slopes is , then the lines are perpendicular.

In this case, the slope of the line $y=-\frac{2}{3}x +7$ is $m=-\frac{2}{3}$ and the slope of the line $y=-\frac{3}{2}x +2$ is $m = -\frac{3}{2}$ .

Since $-\frac{2}{3} \cdot -\frac{3}{2} = 1 eq -1$ , the slopes are not perpendicular.

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Question

Line , which follows the equation , intersects line at . If line also passes through , are and perpendicular?

Answer

The product of perpendicular slopes is always . Knowing this, and seeing that the slope of line is , we know any perpendicular line will have a slope of $m=-\frac{1}{3}$ .

Since line passes through and , we can use the slope equation:

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{13-16}{14-5} = -\frac{3}{9}=-\frac{1}{3}$

Since the two slopes' product is , the lines are perpendicular.

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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

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

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

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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Question

What is the equation for the line that is perpendicular to through point ?

Answer

Perpendicular slopes are opposite reciprocals.

The given slope is found by converting the equation to the slope-intercept form.

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

The slope of the given line is $m = \frac{4}{3}$ and the perpendicular slope is $m = \frac{-3}{4}$ .

We can use the given point and the new slope to find the perpendicular equation. Plug in the slope and the given coordinates to solve for the y-intercept.

$6 = \frac{-3}{4}(4) + b$

Using this y-intercept in slope-intercept form, we get out final equation: $y = \frac{-3}{4}x + 9$ .

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Question

What line is perpendicular to and passes through ?

Answer

Convert the given equation to slope-intercept form.

$y=-\frac{3}{5}+3$

The slope of this line is $-\frac{3}{5}$ . The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.

The perpendicular slope is $\frac{5}{3}$ .

Plug the new slope and the given point into the slope-intercept form to find the y-intercept.

$2 = \frac{5}{3}(3) + b$

So the equation of the perpendicular line is $y = \frac{5}{3}x - 3$ .

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Question

Which of the following is possibly a line perpendicular to ?

Answer

To start, begin by dividing everything by , this will get your equation into the format . This gives you:

$y=\frac{1}{3}x+\frac{22}{9}$

Now, recall that the slope of a perpendicular line is the opposite and reciprocal slope to its mutually perpendicular line. Thus, if our slope is $\frac{1}{3}$ , then the perpendicular line's slope must be . Thus, we need to look at our answers to determine which equation has a slope of . Among the options given, the only one that matches this is . If you solve this for , you will get:

$y = -3x + \frac{18}{3}$

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Question

Which of the following is the equation of a line perpendicular to the line given by:

?

Answer

For two lines to be perpendicular their slopes must have a product of .
$-3*\frac{1}{3} = -1$ and so we see the correct answer is given by $y = \frac{1}{3}x + 2$

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Question

What is the equation of a line perpendicular to the line defined by the equaiton:

Answer

Perpendicular lines have slopes whose product is .

Looking at our equations we can see that it is in slope-intercept form where the m value represents the slope of the line,

.

In our case we see that

therefore, .

Since

$-9*\frac{1}{9} = -1$ we see the only possible answer is

$y = \frac{1}{9}x +6$ .

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