GED Math - Parallel and Perpendicular Lines

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

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

$\small y=2x+4$

$\small y=-2x-2$

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

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

Explanation

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:

$\small y=2x+4$

Line

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?

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

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

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

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

Explanation

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

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

Which of the following lines is perpendicular to the line ?

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

$y=-\frac{4}{3}x-5$

$y=-\frac{1}{4}x-3$

Explanation

Recall that perpendicular lines have slopes that are negative reciprocals.

Start by putting into slope-intercept form.

$y=-\frac{3}{4}x+\frac{5}{8}$

The slope of the given line is $-\frac{3}{4}$ , which means that the line perpendicular to it must have a slope of $\frac{4}{3}$ .

$y=\frac{4}{3}x-2$ is the only line that has the required slope.

Axes_1

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

$- \frac{3}{5}$

$- \frac{5}{3}$

Explanation

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 $- \frac{3}{5}$ .

Axes

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.

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

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

$y = - \frac{1}{3}x -6$

$y = \frac{1}{3}x -6$

Explanation

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:

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

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

$\frac{1}{3}$

$\frac{2}{3}$

$-\frac{1}{3}$

Explanation

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: $\frac{1}{3}$

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

The lines are parallel.

The lines are perpendicular.

The lines are identical.

The lines are distinct but neither parallel nor perpendicular.

Explanation

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 :