Parallel and Perpendicular Lines - GED Math

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Question

Which of the following equations depicts a line that is perpendicular to the line

$\small y=-\frac{1}{2}x+7$ ?

Answer

The given equation is written in slope-intercept form, and the slope of the line is . The slope of a perpendicular line is the negative reciprocal of the given line. The negative reciprocal here is . Therefore, the correct equation is:

$\small y=2x+4$

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Question

Which of the following equations is represented by a line perpendicular to the line ?

Answer

The equation can be rewritten as follows:

$y = -\frac{a}{b}x + \frac{c}{b}$

This is the slope-intercept form, and the line has slope $-\frac{a}{b}$ .

The line therefore has slope $-\frac{3}{5}$ . Since a line perpendicular to this one must have a slope that is the opposite reciprocal of $-\frac{3}{5}$ , we are looking for a line that has slope $\frac{5}{3}$ .

The slopes of the lines in the four choices are as follows:

: $m = -\frac{3}{-5} = \frac{3}{5}$

: $m = -\frac{3}{5}$

: $m = -\frac{5}{3}$

: $m = -\frac{5}{-3} = \frac{5}{3}$ This is the correct choice.

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Question

Which of the following equations is represented by a line perpendicular to the line of the equation $\frac{1}{2}x + \frac{1}{5}y =21$ ?

Answer

The equation can be rewritten as follows:

$y = -\frac{a}{b}x + \frac{c}{b}$

This is the slope-intercept form, and the line has slope $-\frac{a}{b}$ .

The line of the equation $\frac{1}{2}x + \frac{1}{5}y =21$ therefore has slope

$m = -\frac{\frac{1}{2}}{\frac{1}{5}} =- \frac{1}{2} \div \frac{1}{5}= - \frac{1}{2} \cdot \frac{5}{1}= - \frac{5}{2}$

Since a line perpendicular to this one must have a slope that is the opposite reciprocal of $- \frac{5}{2}$ , we are looking for a line with slope $\frac{2}{5}$ .

The slopes of the lines in the four choices are as follows:

; $m = -\frac{5}{-2} = \frac{5}{2}$

; $m = -\frac{5}{2}$

: $m = -\frac{2}{5}$

: $m = -\frac{2}{-5} = \frac{2}{5}$ - this is the correct one.

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Question

Which of the following equations is represented by a line perpendicular to the line of the equation ?

Answer

can be rewritten as follows:

$x= -\frac{13}{4}$

Any line with equation is vertical and has undefined slope; a line perpendicular to this is horizontal and has slope 0, and can be written as . The only choice that does not have an is , which can be rewritten as follows:

$y = - \frac{17}{4}$

This is the correct choice.

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Question

Which of the following equations is represented by a line perpendicular to the line of the equation ?

Answer

The equation can be rewritten as follows:

$y = -\frac{a}{b}x + \frac{c}{b}$

This is the slope-intercept form, and the line has slope $-\frac{a}{b}$ .

The line of the equation has slope

$m = -\frac{0.6}{0.4} =- \frac{3}{2}$

Since a line perpendicular to this one must have a slope that is the opposite reciprocal of $-\frac{3}{2}$ , we are looking for a line that has slope $\frac{2}{3}$ .

The slopes of the lines in the four choices are as follows:

: $m = -\frac{3}{2}$

: $m = -\frac{3}{-2} = \frac{3}{2}$

: $m = -\frac{2}{3}$

: $m = -\frac{2}{-3} = \frac{2}{3}$ - the correct choice.

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Question

Refer to the above red line. A line is drawn perpendicular to that line, and with the same -intercept. What is the equation of that line in slope-intercept form?

Answer

First, we need to find the slope of the above line.

The slope of a line. given two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $x_{1}=-4, y_{1}=x_{2}= 0, y_{2}=8$ :

$m = \frac{8-0}{0-(-4)} = \frac{8}{4} = 2$

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 2, which would be $m = -\frac{1}{2}$ .

Since we want the line to have the same -intercept as the above line, which is the point , we can use the slope-intercept form to help us. We set

$m = -\frac{1}{2},x= -4, y=0$ , and solve for :

$0 = -\frac{1}{2}(-4)+b$

Substitute for and in the slope-intercept form, and the equation is

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

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Question

Refer to the above red line. A line is drawn perpendicular to that line with the same -intercept. Give the equation of that line in slope-intercept form.

Answer

First, we need to find the slope of the above line.

The slope of a line. given two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $x_{1}=-6, y_{1}=x_{2}= 0, y_{2}=18$ :

$m = \frac{18-0}{0-(-6)} = \frac{18}{6} = 3$

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 3, which would be $m = -\frac{1}{3}$ . Since we want this line to have the same -intercept as the first line, which is the point , we can substitute $m = -\frac{1}{3}$ and into the slope-intercept form of the equation:

$y = - \frac{1}{3}x + 18$

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Question

Give the equation of the line parallel to the above red line that includes the origin.

Answer

First, we need to find the slope of the above line.

The slope of a line. given two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $x_{1}=-4, y_{1}=x_{2}= 0, y_{2}=8$ :

$m = \frac{8-0}{0-(-4)} = \frac{8}{4} = 2$

A line parallel to this line also has slope . Since it passes through the origin, its -intercept is , and we can substitute into the slope-intercept form of the equation:

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Question

Consider the equations and . Which of the following statements is true of the lines of these equations?

Answer

We find the slope of each line by putting each equation in slope-intercept form, , and examining the coefficient of .

is already in slope-intercept form; its slope is .

To get in slope-intercept form we solve for :

$\frac{2y }{2}=\frac{-8x+ 13}{2}$

$y=\frac{-8x }{2} + \frac{ 13}{2}$

$y=-4x + \frac{ 13}{2}$

The slope of this line is .

The slopes are not equal so we can eliminate both "parallel" and "identical" as choices.

Multiply the slopes together:

$-4 \times 4 = -16$

The product of the slopes of the lines is not , so we can eliminate "perpendicular" as a choice.

The correct response is "neither".

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Question

Consider the equations and . Which of the following statements is true of the lines of these equations?

Answer

We find the slope of each line by putting each equation in slope-intercept form and examining the coefficient of .

is already in slope-intercept form; its slope is .

To get into slope-intercept form we solve for :

$\frac{9y }{9} =\frac{ - 3x + 10 }{9}$

$y = \frac{-3x}{9}+ \frac{10}{9}$

$y =- \frac{1 }{3}x+ \frac{10}{9}$

The slope of this line is $m = - \frac{1}{3}$ .

The slopes are not equal so we can eliminate both "parallel" and "one and the same" as choices.

Multiply the two slopes together:

$3 \times \left ( - \frac{1}{3} \right ) = -1$

The product of the slopes of the lines is , making the lines perpendicular.

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Question

Line includes the points and . Line includes the points and . Which of the following statements is true of these lines?

Answer

We calculate the slopes of the lines using the slope formula.

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-3 - (-6)}{4-8} = \frac{3}{-4} = - \frac{3}{4}$ .

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-2 - 4}{5- (-3)} = \frac{-6}{8} = - \frac{3}{4}$ .

The lines have the same slope, so either they are distinct parallel lines or one and the same line. One way to check for the latter situation is to find the slope of the line connecting one point on to one point on - if the slope is also $- \frac{3}{4}$ , the lines coincide. We will use and :

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-2 - (-6)}{5-8} = \frac{4}{-3} = - \frac{4}{3}$ .

The lines are therefore distinct and parallel.

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Question

Line includes the points and . Line includes the points and . Which of the following statements is true of these lines?

Answer

We calculate the slopes of the lines using the slope formula.

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-3 - (-6)}{4-8} = \frac{3}{-4} = - \frac{3}{4}$ .

The slope of line is

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{12 - 9}{-16- (-12)} = \frac{3}{-4} = - \frac{3}{4}$ .

The lines have the same slope, so either they are distinct, parallel lines or one and the same line. One way to determine which is the case is to find the equations.

Line , the line through and , has equation

$y - (-3)= -\frac{3}{4} (x - 4)$

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

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

Line , the line through and , has equation

$y - 9= -\frac{3}{4} [x - (-12)]$

$y - 9= -\frac{3}{4} (x +12)$

$y - 9= -\frac{3}{4} x - 9$

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

The lines have the same equation, making them one and the same.

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Question

Give the slope of a line perpendicular to the line in the above figure.

Answer

In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. The slope of a line is the ratio of rise to run, so the slope of the line shown is $\frac{5}{3}$ .

A line perpendicular to this will have a slope equal to the opposite of the reciprocal of $\frac{5}{3}$ . This is $- \frac{3}{5}$ .

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Question

Give ths slope of a line parallel to the line in the above figure.

Answer

In order to move from the lower left plotted point to the upper right plotted point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. The slope is the rise-to-run ratio, so the slope of the line is $\frac{5}{3}$ . Any line parallel to this line will have the same slope, so the correct response is $\frac{5}{3}$ .

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Question

Given the following equation, what is the slope of the perpendicular line?

Answer

Subtract from both sides.

The slope of this line is negative three.

The slope of the perpendicular line is the negative reciprocal of this slope.

$m_{\perp} = -\frac{1}{m} = -\frac{1}{-3} = \frac{1}{3}$

The answer is: $\frac{1}{3}$

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Question

Which line is parallel to the following:

Answer

Two lines are parallel if they have the same slope. Now, we know the slope-intercept form is written as follows:

where m is the slope and b is the y-intercept. Now, given the equation

we can see the slope is -3. So, to find a line parallel to this line, we will have to find an equation that also have a slope of -3.

If we look at the equation

we can see it has a slope of -3. Therefore, this equation is parallel to the original equation.

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Question

Find a line that is perpendicular to the following:

$y = \frac{1}{2}x-9$

Answer

Two lines are perpendicular if their slopes have opposite signs (positive/negative) and they are reciprocals of each other.

To find a reciprocal of a number, we will write it in fraction form. Then, the numerator becomes the denominator and the denominator becomes the numerator. In other words, we flip the fraction.

We will look at the lines in slope-intercept form

where m is the slope and b is the y-intercept.

So, given the equation

$y = \frac{1}{2}x-9$

we can see the slope is $\frac{1}{2}$ . Now, the opposite reciprocal of this slope is $-\frac{2}{1}$ which is the same as . So, we will find the equation that has as the slope.

So, in the equation

we can see the slope is . Therefore, it is perpendicular to the original equation.

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Question

Find a line that is parallel to the following line:

Answer

Two lines are parallel if they have the same slope. So, we will look at the lines in slope-intercept form:

where m is the slope and b is the y-intercept.

So, given the line

we can see the slope is -2. So, to find a line that is parallel, it must also have a slope of -2. So, the line

we can see it also has a slope of -2. Therefore, it is parallel to the original line.

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Question

If a line has a slope of $-\frac{2}{3}$ , what must be the slope of the perpendicular line?

Answer

The perpendicular line slope will be the negative reciprocal of the original slope.

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

Substitute the given slope into the equation.

$m_{\perp} = -\frac{1}{-\frac{2}{3}} = 1\div \frac{2}{3} = 1\times \frac{3}{2}$

The answer is: $\frac{3}{2}$

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Question

If the slope of a line is $-\frac{4}{3}$ , what must be the slope of the perpendicular line?

Answer

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

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

Substitute the slope into the equation.

$m_{\perp} = -\frac{1}{(-\frac{4}{3})} = 1 \div \frac{4}{3} = 1\times \frac{3}{4}$

The answer is: $\frac{3}{4}$

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