Perpendicular Lines

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ACT Math › Perpendicular Lines

Questions 1 - 10

Which of the following lines is perpendicular to the line ?

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

Explanation

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.

What line is perpendicular to and passes through ?

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

$y = \frac{-3}{5}x + 2$

$y = \frac{1}{3}x - 5$

$y = \frac{2}{5}x + 4$

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

Explanation

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 $y = \frac{5}{3}x - 3$ .

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

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

Explanation

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

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

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

Explanation

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

What line is perpendicular to and passes through ?

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

$y = \frac{-3}{5}x + 2$

$y = \frac{1}{3}x - 5$

$y = \frac{2}{5}x + 4$

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

Explanation

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 $y = \frac{5}{3}x - 3$ .

Which of the following lines is perpendicular to the line ?

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

Explanation

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.

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

$y = -\frac{1}{3}x + 5$

$y = \frac{1}{3}x -7$

Explanation

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:

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

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

$y = -\frac{1}{3}x + 5$

$y = \frac{1}{3}x -7$

Explanation

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:

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

What is the slope of a line that is perpendicular to the equation given by:

$m = -\frac{1}{6}$

$m = \frac{1}{6}$

Explanation

Perependicular lines have slopes whose product is .

$6*(-\frac{1}{6}) = -1$ and so the answer is
$m = -\frac{1}{6}$

Calculate the slope of a line perpendicular to the line with the following equation:

$\frac{1}{3}$

$-\frac{1}{3}$

None of these

Explanation

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 .

First let's find the negative of the current slope.

$-3\rightarrow -(-3) \textup{ or }3$

Now, we need to find the reciprocal of . In order to find the reciprocal of a number we divide one by that number; therefore, we can calculate the following:

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

The negative reciprocal will be $+\frac{1}{3}$ or $\frac{1}{3}$ which will be the slope of the perpendicular line.

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