How to find the slope of perpendicular lines - Algebra

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Question

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

Answer

Lines can be written in the slope-intercept form:

In this equation, is the slope and is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

The reciprocal of is ; therefore, the reciprocal of is .

Next, change the sign.

$-1\rightarrow 1$

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Question

Find the slope of the line that is perpendicular to the line that contains (1, 9) and (3, 4).

Answer

The slope of the line that contains the points (1, 9) and (3, 4) is $\frac{4-9}{3-1}=\frac{-5}{2}$ .

$slope=\frac{y_2-y_1}{x_2-x_1}=\frac{4-9}{3-1}=-\frac{5}{2}$

The negative reciprocal is $\frac{2}{5}$ , which is the slope of the perpendicular line.

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Question

What is the slope of a line perpendicular to 10x + 4y = 20?

Answer

First, you should put the equation in slope intercept form (y = mx + b), where m is the slope.

10x + 4y = 20

Isolate the y term

10x + 4y – 10x = 20 – 10x

4y = 20 – 10x

Rearrange the terms

4y = –10x + 20

Divide both sides by 4

$\frac{4y}{4}=\frac{-10x+20}{4}$

$y=-\frac{10}{4}x+5$

The slope of the given line is –10/4. A perpendicular line will have a slope with the opposite reciprocal. In simpler terms, you flip the slope and change the sign, therefore, the opposite reciprocal is 4/10.

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Question

A line passes through points (–6,9) and (4,–4). Find the slope of the line that runs perpendicular to this line.

Answer

To find the slope of this perpendicular line, we need to first find the slope of the line that passes through points (–6,9) and (4,–4). Remember, the general formula for slope is $m=\frac{y_2-y_1}{x_2-x_1}$ , where the two points are and . In our case, we can calculate the slope using out two points.

$m=\frac{-4-9}{4-(-6)}=-\frac{13}{10}$

The slope of the line passing through (–6,9) and (4,–4) is –13/10. To find the slope of the line that is perpendicular, just switch the sign and take the reciprocal of –13/10. This gives us 10/13. So 10/13 is the slope of that perpendicular line.

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Question

If the slope of a line is equal to $\frac{1}{2}$ , what is the slope of its perpendicular intercept?

Answer

Slope of lines that are perpendicular to each other are the negative reciprocal.

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

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Question

Two lines are perpendicular to each other. One of the lines is depicted by the equation . What is the slope of the other line?

Answer

Perpendicular lines have slopes that are negative reciprocals of one another. Since the original line has a slope of , the perpendicular line must have a slope of $-\frac{1}{5}$ .

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Question

Two lines are perpendicular to each other. One of the lines' equations is .

What is the slope of the other line?

Answer

Perpendicular lines have slopes that are negative reciprocals of one another. The given line's slope is 5, which means that the slope of the other line must be its negative reciprocal. The negative reciprocal of 5 is $-\frac{1}{5}$ .

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Question

What is the slope of a line that is perpendicular to

$y=\frac{6}{5}x-9$ ?

Answer

The slopes of perpendicular lines are negative inverses of each other. The slope of the given line is $\frac{6}{5}$ . The negative inverse of $\frac{6}{5}$ is $-\frac{5}{6}$ .

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Question

What is the slope of a line that is perpendicular to $y=\frac{2}{3}x-6$

Answer

Perpendicular lines have slopes that are negative inverses of one another. The slope of the given line is $\frac{2}{3}$ . The negative inverse of $\frac{2}{3}$ is $-\frac{3}{2}$ , which must be the slope of the perpendicular line.

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Question

Find the slope of a line that's perpendicular to the following linear equation:

Answer

We are given

To find the slope that's perpendicular, we perform the following steps

First, take the slope from our original equation. In our equation, the slope is

Take the reciprocal of that slope. The reciprocal is $-\frac{1}{2}$ .

Finally, change the sign so that we end up with $\frac{1}{2}$ . This is the number that represents the slope of the perpendicular line.

Another way to think of this problem is that the general formula for the slope that's perpendicular is

$slope_\perp=-\frac{1}{m}$

where is the slope of the original equation. In our case, . Thus,

$slope_\perp=-\frac{1}{-2}=\frac{1}{2}$

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Question

Find the slope of the line perpendicular to the line that fits the following points:

(3,5), (2,7), (0,11)

Answer

To find the slope of the perpendicular line, we must first find the slope of the line fitting the given points. Slope is equal to change in over change in .

$\frac{7-5}{2-3}=\frac{2}{-1}=-2$

The perpendicular slope is the opposite reciprocal of the slope of the line to which it is perpendicular. So flip the original slope upside down and multiply by .

Perpendicular slope

$=\frac{1}{2}$

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Question

What is the slope of a line that is perpendicular to

$y=\frac{3}{2}x+9\ ?$

Answer

Perpendicular lines have slopes that are negative reciprocals of one another. The slope of the given line is $\frac{3}{2}$ , which has a negative reciprocal of $-\frac{2}{3}$ .

Thus, the slope of the perpendicular line must be $-\frac{2}{3}$ .

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Question

What is the slope of a line which is perpendicular to the following line?

$y=\frac{5}{9}x+8$

Answer

Given that our equation is in slop-intercept form

$y=\frac{5}{9}x+8$

where is the slope, we see that for this line the slope is $\frac{5}{9}$ .

Find the negative reciprocal of the slope to find the perpendicular line's slope.

Flip the fraction and make it negative.

$\frac{5}{9}\rightarrow -\frac{9}{5}$ .

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Question

$\ \textup{The slope of a side of a square is 3.}\\textup{Which of these values cannot be the slope of one of the other three sides?}$

Answer

$\textup{Of the other three sides, the opposite side lays on a parallel line,}\\textup{ so its slope is 3 as well. The two adjacent sides lay on perpendicular lines,}\\textup{ so their slope is the inverse reciprocal of 3; that is, } -\frac{1}{3}.$

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Question

$\\textup{Two opposite vertices of a square are the points (0,0) and (-2,4).}\\textup{Which can be one of the other two vertices?}$

Answer

$\textup{The segment between the two opposite vertices (0,0) and (-2,4) is one }\\textup{diagonal of the square with slope}\frac{4}{-2}=-2 \textup{ and midpoint } \left(\frac{0+(-2)}{2},\frac{0+4}{2} \right)=(-1,2). \textup{ The other diagonal lays on the line } y=\frac{1}{2}x+p \textup{ since it is perpendicular}\\textup{ and has as a slope of the inverse reciprocal. Substituting the midpoint into the }\\textup{ parametric equation we obtain } p=\frac{5}{2}\textup{ and the line equation } y=\frac{1}{2}c+\frac{5}{2}.$

$\textup{ Only point } (1,3) \textup{ among the possible answers lays on this line.}$ .

$\textup{ Alternatively, one can check for every possible answer if the two segments}\\textup{between the candidate point and the two opposite vertices are perpendicular.}$

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Question

What is the slope of a line perpendicular to $y=-\frac{1}{2}x-5$ ?

Answer

Perpendicular lines are lines that intersect each other at a ninety degree angle. The slope of a line that is perpendicular to another has the opposite sign and is the reciprocal. The slope of a line perpendicular to the one given would be .

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Question

Find the slope of the line perpendicular to .

Answer

The slope of a perpendicular line is always the negative reciprocal of the original slope.

Our original equation is which is in the form where is the slope. Therefore, the original slope is .

To find the slope of a line perpendicular we want to use the following formula,

$m_p=-\frac{1}{m}$

$m_p=-\frac{1}{3}$ .

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Question

Find the slope of the line that is perpendicular to a line with the equation:

Answer

Lines can be written in the slope-intercept form:

In this equation, is the slope and is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of :

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

Flip the numerator and the denominator.

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

Next, change the sign.

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

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Question

Find the slope of the line that is perpendicular to a line with the equation:

$y=\frac{1}{3}x-9$

Answer

Lines can be written in the slope-intercept form:

In this equation, is the slope and is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of $\frac{1}{3}$ .

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

Next, change the sign.

$3\rightarrow-3$

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Question

Find the slope of the line that is perpendicular to a line with the equation:

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

Answer

Lines can be written in the slope-intercept form:

In this equation, is the slope and is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of $-\frac{2}{3}$ .

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

Next, change the sign.

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

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