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GED Math : Geometry and Graphs

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Example Questions

Example Question #4 : Parallel And Perpendicular Lines

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.

Possible Answers:

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

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

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

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

Correct answer:

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

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 (0,18) , we can substitute $m = -\frac{1}{3}$ and b = 18 into the slope-intercept form of the equation:

y = mx + b

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

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Example Question #4 : Parallel And Perpendicular Lines

Line includes the points (4, -3) and (8, -6) . Line includes the points (5, -2) and (-3, 4) . Which of the following statements is true of these lines?

Possible Answers:

The lines are distinct but neither parallel nor perpendicular.

The lines are identical.

The lines are perpendicular.

The lines are parallel.

Correct answer:

The lines are parallel.

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 (8, -6) and (5, -2) :

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

The lines are therefore distinct and parallel.

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Example Question #3 : Parallel And Perpendicular Lines

Line includes the points (4, -3) and (8, -6) . Line includes the points (-12, 9) and (-16, 12) . Which of the following statements is true of these lines?

Possible Answers:

The lines are perpendicular.

The lines are identical.

The lines are distinct but neither parallel nor perpendicular.

The lines are parallel.

Correct answer:

The lines are identical.

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

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

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 (4, -3) and (8, -6) , 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 (-12, 9) and (-16, 12) , 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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Example Question #1 : Parallel And Perpendicular Lines

Line

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

Possible Answers:

y = 2x

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

y = -2x

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

Correct answer:

y = 2x

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

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

A line parallel to this line also has slope m = 2 . Since it passes through the origin, its -intercept is (0,0) , and we can substitute m = 2,b=0 into the slope-intercept form of the equation:

y = mx+b

y = 2x + 0

y = 2x

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Example Question #3 : Parallel And Perpendicular Lines

Consider the equations y = 4x + 9 and 8x + 2y = 13 . Which of the following statements is true of the lines of these equations?

Possible Answers:

The lines are perpendicular.

The lines are parallel.

The lines are distinct but neither parallel nor perpendicular.

The lines are identical.

Correct answer:

The lines are distinct but neither parallel nor perpendicular.

Explanation:

We find the slope of each line by putting each equation in slope-intercept form, y =mx + b , and examining the coefficient of .

y = 4x + 9 is already in slope-intercept form; its slope is m = 4 .

To get 8x + 2y = 13 in slope-intercept form we solve for :

8x + 2y = 13

8x + 2y -8x = 13 -8x

2y =-8x+ 13

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

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

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

The slope of this line is m =-4 .

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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Example Question #2 : Parallel And Perpendicular Lines

Consider the equations y = 3x + 7 and 3x + 9y = 10 . Which of the following statements is true of the lines of these equations?

Possible Answers:

The lines are parallel.

The lines are perpendicular.

The lines are distinct but neither parallel nor perpendicular.

The lines are one and the same.

Correct answer:

The lines are perpendicular.

Explanation:

We find the slope of each line by putting each equation in slope-intercept form y =mx + b and examining the coefficient of .

y = 3x + 7 is already in slope-intercept form; its slope is m = 3 .

To get 3x + 9y = 10 into slope-intercept form we solve for :

3x + 9y = 10

3x + 9y - 3x = 10 - 3x

9y = - 3x + 10

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

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

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

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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Example Question #771 : Geometry And Graphs

Axes_1

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

Possible Answers:

$- \frac{5}{3}$

$\frac{5}{3}$

$\frac{3}{5}$

$- \frac{3}{5}$

Correct answer:

$\frac{5}{3}$

Explanation:

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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Example Question #11 : Parallel And Perpendicular Lines

Axes_1

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

Possible Answers:

$- \frac{5}{3}$

$- \frac{3}{5}$

Correct answer:

$- \frac{3}{5}$

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

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Example Question #11 : Parallel And Perpendicular Lines

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?

Possible Answers:

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

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

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

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

Correct answer:

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

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 (-4,0) , we can use the slope-intercept form to help us. We set

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

y = mx+b

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

0 = 2 + b

b = -2

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

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

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Example Question #13 : Parallel And Perpendicular Lines

Which of the following equations is represented by a line perpendicular to the line 3x + 5y =21 ?

Possible Answers:

3x - 5y =19

3x + 5y =19

5x - 3y = 19

5x + 3y = 19

Correct answer:

5x - 3y = 19

Explanation:

The equation ax + by = c can be rewritten as follows:

ax + by = c

by = -ax + c

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

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

The line 3x + 5y =21 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:

3x - 5y =19 : $m = -\frac{3}{-5} = \frac{3}{5}$

3x + 5y =19 : $m = -\frac{3}{5}$

5x + 3y = 19 : $m = -\frac{5}{3}$

5x - 3y = 19 : $m = -\frac{5}{-3} = \frac{5}{3}$ This is the correct choice.

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GED Math : Geometry and Graphs

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #4 : Parallel And Perpendicular Lines

Example Question #4 : Parallel And Perpendicular Lines

Example Question #3 : Parallel And Perpendicular Lines

Example Question #1 : Parallel And Perpendicular Lines

Example Question #3 : Parallel And Perpendicular Lines

Example Question #2 : Parallel And Perpendicular Lines

Example Question #771 : Geometry And Graphs

Example Question #11 : Parallel And Perpendicular Lines

Example Question #11 : Parallel And Perpendicular Lines

Example Question #13 : Parallel And Perpendicular Lines

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