Perpendicular Lines - Math

Card 0 of 76

Question

Find the equation of a line perpendicular to

Answer

Since a perpendicular line has a slope that is the negative reciprocal of the original line, the new slope is $\frac{1}{3}$ . There is only one answer with the correct slope.

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Question

What is the equation, in slope-intercept form, of the perpendicular bisector of the line segment that connects the points $\small (2,2)$ and $\small (8,6)$ ?

Answer

First, calculate the slope of the line segment between the given points.

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-2}{8-2}=\frac{4}{6}=\frac{2}{3}$

We want a line that is perpendicular to this segment and passes through its midpoint. The slope of a perpendicular line is the negative inverse. The slope of the perpendicular bisector will be $m=-\frac{3}{2}$ .

Next, we need to find the midpoint of the segment, using the midpoint formula.

$mid=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{x})=(\frac{2+8}{2},\frac{2+6}{2})=(5,4)$

Using the midpoint and the slope, we can solve for the value of the y-intercept.

$4=(-\frac{3}{2})(5)+b$

$4=\frac{8}{2}=-\frac{15}{2}+b$

$\frac{23}{2}=b$

Using this value, we can write the equation for the perpendicular bisector in slope-intercept form.

$y=-\frac{3}{2}x+\frac{23}{2}$

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Question

Write an equation in slope-intercept form for the line that passes through $\left ( 3,1 \right )$ and that is perpendicular to a line which passes through the two points $\left ( 3,1 \right )$ and $\left ( -2,5 \right )$ .

Answer

Find the slope of the line through the two points. It is $-\frac{4}{5}$ .

Since the slope of a perpendicular line is the negative reciprocal of the original line, the new line's slope is $\frac{5}{4}$ . Plug the slope and one of the points into the point-slope formula $y-y_{1}=m(x-x_{1})$ . Isolate for .

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Question

Which of the following are perpendicular to the line with the formula $y = -\frac{1}{3}X + 3$ ?

I.

II. $y = \frac{1}{3}X + 3$

III. $y = \frac{4.5}{1.5}X -3$

Answer

The slope of a perpendicular line is equal to the negative reciprocal of the original line. This means that the slope of our perpendicular line must be 3. We can also note that $\frac{4.5}{1.5}$ is also equal to 3, so both of these slopes are correct. The y-intercept does not matter, as the slope is the only thing that determines the slant of the line. Therefore, numerals I and III are both correct.

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Question

Which of the following lines is perpendicular to ?

Answer

In order for two lines to be perpendicular to each other, their slopes must be opposites and reciprocals of each other, meaning the fraction must be flipped upside down and the signs must be changed. In this situation, the original equation had a slope of , so the perpendicular slope must be $-\frac{1}{2}$ .

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Question

Which of the following lines will be perpendicular to ?

Answer

Two lines are perpendicular if they have opposite reciprocal slopes. When a line is in standard form, the is the slope. A perpendicular line will have a slope of $-\frac{1}{m}$ .

The slope of our given line is . Therefore we want a slope of $-\frac{1}{2}$ . The only line with the correct slope is $y=-\frac{1}{2}x+1$ .

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Question

Find the equation (in slope-intercept form) of a line perpendicular to .

Answer

First, find the slope of the original line, which is $-\frac{3}{2}$ . You can do this by isolating for so that the equation is in slope-intercept form. Once you find the slope, just replace the in the original equation withe the negative reciprocal (perpendicular lines have a negative reciprocal slope for each other). Thus, your answer is

$y=\frac{2}{3}X+\frac{11}{2}$

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Question

Given the equation and the point , find the equation of a line that is perpendicular to the original line and passes through the given point.

Answer

In order for two lines to be perpendicular, their slopes must be opposites and recipricals of each other. The first step is to find the slope of the given equation:

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

Therefore, the slope of the perpendicular line must be . Using the point-slope formula, we can find the equation of the new line:

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Question

What line is perpendicular to $y=\frac{1}{2}x+3$ through ?

Answer

Perdendicular lines have slopes that are opposite reciprocals. The slope of the old line is $\frac{1}{2}$ , so the new slope is .

Plug the new slope and the given point into the slope intercept equation to calculate the intercept:

or , so .

Thus , or .

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Question

What line is perpendicular to through ?

Answer

The equation is given in the slope-intercept form, so we know the slope is . To have perpendicular lines, the new slope must be the opposite reciprocal of the old slope, or $\frac{-1}{2}$

Then plug the new slope and the point into the slope-intercept form of the equation:

$-2=\frac{-1}{2}(6)+b$ so so

So the new equation becomes: $y=\frac{-1}{2}x+1$ and in standard form

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Question

A line passes through the points and . If a new line is drawn perpendicular to the original line, what will its slope be?

Answer

The original line has a slope of $\frac{11}{3}$ , a line perpendicular to the original line will have a slope which is the negative reciprocal of this value.

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Question

Find the slope of the line perpendicular to $y + \frac{2}{3}x = 12$ .

Answer

Put this equation into slope-intercept form, y = mx + b, to find the slope, m.

Do this by subtracting $\frac{2}{3}x$ from both sides of the equation:

$y = -\frac{2}{3}x + 12$

The slope of this line is $-\frac{2}{3}$ .

The slope of the perpendicular line is the negative reciprocal. Switch the numerator and denominator, and then multiply by :

$-\frac{2}{3} \Rightarrow -\frac{3}{2} \Rightarrow -\frac{3}{2} \times -1 \Rightarrow \frac{3}{2}$

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Question

Which of the following is the equation of a line that is perpendicular to the line ?

Answer

Perpendicular lines have slopes that are the opposite reciprocals of each other. Thus, we first identify the slope of the given line, which is (since it is in the form , where represents slope).

Then, we know that any line which is perpendicular to this will have a slope of $-\frac{1}{3}$ .

Thus, we can determine that $y = -\frac{1}{3}x + 10$ is the only choice with the correct slope.

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Question

Find the slope of this line:

Answer

Isolate for so that the equation now reads

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

The slope is $\frac{1}{4}$ .

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Question

Which of the following is perpendicular to the line described by

Answer

The definition of perpendicular lines is that their slopes are inverse reciprocals of one another. Since the slope in the given equation is , this means that the slope of its perpendicular line would be $-\frac{1}{3}$ .

The answer

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

is the only equation listed that has a slope of $-\frac{1}{3}$ .

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Question

Which of the following is perpendicular to the line described by

Answer

The definition of perpendicular lines is that their slopes are inverse reciprocals of one another. Since the slope in the given equation is , this means that the slope of its perpendicular line would be $\frac{1}{6}$ .

The answer

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

is the only equation listed that has a slope of $\frac{1}{6}$ .

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Question

What will be the slope of the line perpendicular to $y=3x+\frac{1}{2}$ ?

Answer

In standard form, the is the slope.

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

For our given line, the slope is . Therefore, the slope of the perpendicular line is $-\frac{1}{3}$ .

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Question

What is the slope of the line perpendicular to $y=\frac{2}{3}x+\frac{1}{2}$ ?

Answer

In standard form, is the slope.

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

For our given line, the slope is $\frac{2}{3}$ . Therefore, the slope of the perpendicular line is $-\frac{3}{2}$ .

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Question

Which of the following gives the slope of a line that is perpendicular to ?

Answer

Recall that the slopes of perpendicular lines are opposite reciprocals of one another. As a result, we are looking for the opposite reciprocal of . Thus, we can get that the opposite reciprocal is $\frac{1}{3}$ .

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Question

Which of the following are perpendicular to the line with the formula $y = -\frac{1}{3}X + 3$ ?

I.

II. $y = \frac{1}{3}X + 3$

III. $y = \frac{4.5}{1.5}X -3$

Answer

The slope of a perpendicular line is equal to the negative reciprocal of the original line. This means that the slope of our perpendicular line must be 3. We can also note that $\frac{4.5}{1.5}$ is also equal to 3, so both of these slopes are correct. The y-intercept does not matter, as the slope is the only thing that determines the slant of the line. Therefore, numerals I and III are both correct.

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