GED Math : Parallel and Perpendicular Lines

Study concepts, example questions & explanations for GED Math

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Example Questions

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Example Question #1 : Parallel And Perpendicular Lines

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

?

Possible Answers:

Correct answer:

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:

Example Question #2 : Parallel And Perpendicular Lines

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

Possible Answers:

Correct answer:

Explanation:

The equation  can be rewritten as follows:

This is the slope-intercept form, and the line has slope 

The line of the equation  therefore has slope

Since a line perpendicular to this one must have a slope that is the opposite reciprocal of , we are looking for a line with slope .

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

 - this is the correct one.

Example Question #3 : Parallel And Perpendicular Lines

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

Possible Answers:

Correct answer:

Explanation:

 can be rewritten as follows:

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:

This is the correct choice.

Example Question #4 : Parallel And Perpendicular Lines

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

Possible Answers:

Correct answer:

Explanation:

The equation  can be rewritten as follows:

This is the slope-intercept form, and the line has slope 

The line of the equation  has slope

Since a line perpendicular to this one must have a slope that is the opposite reciprocal of  , we are looking for a line that has slope .

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

 - the correct choice.

Example Question #5 : 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:

Correct answer:

Explanation:

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

The slope of a line. given two points  can be calculated using the slope formula:

Set :

 

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

Example Question #6 : Parallel And Perpendicular Lines

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

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 parallel.

Explanation:

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

The slope of line  is 

.

The slope of line  is 

.

 

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 , the lines coincide. We will use  and :

.

The lines are therefore distinct and parallel.

Example Question #7 : Parallel And Perpendicular Lines

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

Possible Answers:

The lines are perpendicular.

The lines are distinct but neither parallel nor perpendicular.

The lines are identical.

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 

.

The slope of line  is 

.

 

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

 

Line , the line through  and , has equation

 

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

Example Question #8 : Parallel And Perpendicular Lines

Line

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

Possible Answers:

Correct answer:

Explanation:

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

The slope of a line. given two points  can be calculated using the slope formula:

Set :

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:

Example Question #9 : Parallel And Perpendicular Lines

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

Possible Answers:

The lines are identical.

The lines are perpendicular.

The lines are parallel.

The lines are distinct but neither parallel nor perpendicular.

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, , and examining the coefficient of .

 

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

 

To get  in slope-intercept form we solve for :

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:

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

The correct response is "neither".

Example Question #10 : Parallel And Perpendicular Lines

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

Possible Answers:

The lines are distinct but neither parallel nor perpendicular.

The lines are perpendicular.

The lines are one and the same.

The lines are parallel.

Correct answer:

The lines are perpendicular.

Explanation:

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 :

The slope of this line is .

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

Multiply the two slopes together:

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

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