Algebra 1 : How to find the equation of a perpendicular line

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #11 : How To Find The Equation Of A Perpendicular Line

Write the equation of a line passing through the point  that is perpendicular to the line .

Possible Answers:

Correct answer:

Explanation:

To solve this type of problem, we have to be familiar with the slope-intercept form of a line, where m is the slope and b is the y-intercept. The line that our line is perpendicular to has the slope-intercept equation , which means that the slope is .

The slope of a perpendicular line would be the negative reciprocal, so our slope is .

We don't know the y-intercept of our line yet, so we can only write the equation as:

.

We do know that the point is on this line, so to solve for b we can plug in -2 for x and 3 for y:

First we can multiply to get .

This makes our equation now:

either by subtracting 3 from both sides, or just by looking at this critically, we can see that b = 0.

Our original becomes , or simply .

Example Question #81 : Perpendicular Lines

Write the equation of a line perpendicular to 

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

Two lines are perpendicular if and only if their slopes are "negative reciprocals" of each other such as 1/2 and -2. In our problem we are not given line equations that we can readily see the slope, so we must convert each equation to slope intercept form or .

First find out the slope of the given equation by converting it to slope-intercept form:

 

So we need a line whose slope is the negative reciprocal of -1/8 (which is 8). Even though this number is not "negative" the idea of the negative reicprocal gives a positive number here because two negative signs cancel each other to make a positive. ...negative recicprocal....

Now we must choose an equation that, after being changed to slope intercept form has a slope of 8.

 

So   (answer) is the equation of a line that is perpendicular to 

.

Example Question #83 : Perpendicular Lines

What is a line that is perpendicular to ?

Possible Answers:

Correct answer:

Explanation:

A line is perpendicular to another line when they meet at a degree angle. That angle is the result of the slopes of the lines being opposite reciprocals.

The "opposite reciprocal" of  is best described as .

First we reorganize the original equation to isolate . To do this we want to get our equation into slope intercept form .

First subtract 12 from each side.

Now divide by 4 and simplify where possible.

opposite reciprocal

The only equation in the answer choices with a slope of is

.

Example Question #84 : Perpendicular Lines

A line perpendicular to  passes through the points  and . Find the equation of this line. 

Possible Answers:

Correct answer:

Explanation:

This problem can be quickly solved through using the point-slope formula considering the given information. Before we start substituting in values, however, it's important to remember what determines a line to be perpendicular relative to another. By definition, lines are perpendicular if their slopes have a product of . For instance, if a line has a slope of , the line perpendicular to it will have a slope of  because . Using this concept, we must first determine the slope of the perpendicular line. We are given that the reference line has a slope of . That means the perpendicular line must have a slope of  because . Now that we know  (slope) and have a coordinate, we have fulfilled the requirements of the point-slope formula and can begin to substitue in information and solve for the equation. 

Here we arbitrarily use 

Example Question #85 : Perpendicular Lines

A line has a y-intercept of 7 and is perpendicular to a line with a slope of  . What is the equation of this perpendicular line?

Possible Answers:

Correct answer:

Explanation:

Before beginning this problem, it's important to remember what defines a perpendicular line. A line is perpendicular to another if the product of the two lines' slope is  For instance, if a line has a slope of , the line perpendicular to it will have a slope of  because . Using this concept, we must first determine the slope of the perpendicular line. The line of interest is perpendicular to a line with a slope of , therefore its slope must be 

The problem does not provide us enough information to use the point-slope formula to solve for the equation. However, we are provided with the line's y-intercept. This allows us to use the skeleton  to solve for the equation, where  is slope and  is the y-intercept.

Example Question #86 : Perpendicular Lines

A line is perpendicular to  and has a y-intercept of . Find the equation of this perpendicular line. 

Possible Answers:

Correct answer:

Explanation:

By definition, lines are perpendicular if the product of their slopes is . For example, if a line has a slope of , the perpendicular line would have a slope of . This same concept can be used to solve this problem. Beginning with the issue of slope, we realize that the reference slope is . In order to create a product of , we must multiply  by . Therefore, the slope of the perpendicular line is

Because the problem has not given us a point, we cannot use the point-slope formula to solve for the equation. However, we have been provided with the line's y-intercept. With this information, we are ready to construct the line's equation remembering the  skeleton. We have  and now we have . Substituting in the given information will yield our answer. 

Because  is usally not written in front of variables, we may omit it from the final answer. 

Example Question #331 : Functions And Lines

Line 1 passes through points (0,2) and (3,3). Line 2 is perpindicular to Line 1. Also, Line 2 passes through the point (8,5). Which of the following represents the equation of Line 2?

Possible Answers:

Correct answer:

Explanation:

Begin by calculating the slope of Line 1 as 

.

Now, realize that since Line 2 is perpindicular to Line 1, the slope of Line 2 is the negative reciprocal of the slope of Line 1.

Therefore, the slope of Line 2 is given as 

.

We also know that Line 2 passes through point (8,5).

Therefore, we may use "point-slope" form to express the equation of Line 2 as: 

.

Finally, convert this "point-slope" equation to "slope-intercept" form in order to match the result with the correct answer choice: 

Example Question #82 : Perpendicular Lines

What is the equation of a line that is perpendicular to  and passes through point ?

Possible Answers:

Correct answer:

Explanation:

When finding the slope of a perpendicular line, we need to ensure we have  form.    stands for slope. Our  is . To find the perpendicular slope, we need to take the negative reciprocal of that value which is . Since we are looking for an equation, we need to reuse the  form to solve for  . We do this by plugging in our coordinates. 

 

 Add  on both sides.

 

Our equation is now .

Example Question #89 : Perpendicular Lines

What's the equation of a line that is perpendicular to  and passes through point ?

Possible Answers:

Correct answer:

Explanation:

When finding the slope of a perpendicular line, we need to ensure we have  form.  Let's rearrange it. By subtracting  on both sides and dividing  on both sides, we get  stands for slope. Our  is . To find the perpendicular slope, we need to take the negative reciprocal of that value which is . Since we are looking for an equation, we need to reuse the  form to solve for  . We do this by plugging in our coordinates. 

 

 Subtract  on both sides.

 

Our equation is now .

Example Question #90 : Perpendicular Lines

What's the equation of a line that is perpendicular to  and goes through point ?

Possible Answers:

Correct answer:

Explanation:

When finding the slope of a perpendicular line, we need to ensure we have  form.  Let's rearrange it. By add  on both sides and dividing  on both sides, we get  stands for slope. Our  is . To find the perpendicular slope, we need to take the negative reciprocal of that value which is . Since we are looking for an equation, we need to reuse the  form to solve for  . We do this by plugging in our coordinates. 

 

 Add  on both sides.

 

Our equation is now .

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