# Intermediate Geometry : How to find out if lines are perpendicular

## Example Questions

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### Example Question #1 : How To Find Out If Lines Are Perpendicular

Which of the following lines is perpendicular to the line

Explanation:

The slope of the given line is 3. The slope of a perpendicular line is the negative inverse of the given line. In this case, that is equal to . Therefore, the correct answer is:

### Example Question #1 : How To Find Out If Lines Are Perpendicular

Given the equation of a line:

Which equation given below is perpendicular to the given line?

Explanation:

When looking at the equation of the given line, we know that the slope is  and the y-intercept is . Any line perpendicular to the given line will create a 90 degree angle with the given line.

Now, there are INFINITE lines perpendicular to the given line, all with different y-intercepts. So in other words, the line we are looking for will have no dependence on the y-intercept, as any y-intercept will do. What we do care about is the slope of the line.

The slope of any line perpendicular to a given line has a negative reciprocal of the slope. So for this problem:

The only line that has this slope is

### Example Question #3 : How To Find Out If Lines Are Perpendicular

Given the equation of a line:

Find the equation of a line parallel to the given line.

Explanation:

Parallel lines will never touch, and therefore they must have the same slope.

Many of the answers are reciprocals or negative slopes, but the slope we are looking for is .

That leaves us with 2 answers. However, one of the answers is the exact same equation for a line as the given equation. Therefore our answer is:

### Example Question #4 : How To Find Out If Lines Are Perpendicular

Which of the following is perpendicular to

Explanation:

Two lines are perpendicular if and only if their slopes are negative reciprocals. To find the slope, we must put the equation into slope-intercept form,  , where  equals the slope of the line. We begin by subtracting  from each side, giving us . Next, we subtract 32 from each side, giving us . Finally, we divide each side by , giving us . We can now see that the slope is . Therefore, any line perpendicular to  must have a slope of . Of the equations above, only  has a slope of .

### Example Question #5 : How To Find Out If Lines Are Perpendicular

Which of the following equations is perpendicular to ?

Explanation:

Convert the given equation to slope-intercept form:

Divide both sides of the equation by :

The slope of this function is :

The slope of the perpendicular line will be the negative reciprocal of the original slope. Substitute and solve:

Only  has a slope of .

### Example Question #6 : How To Find Out If Lines Are Perpendicular

Which line is perpendicular to the given line below?

Explanation:

Two perpendicular lines have slopes that are opposite reciprocals, meaning that the sign changes and the reciprocal of the slope is taken.

The original equation is in slope-intercept form,

where  represents the slope.

In this case, the slope of the original is:

After taking the opposite reciprocal, the result is the slope below:

### Example Question #1 : How To Find Out If Lines Are Perpendicular

Are the lines of the equations

and

parallel, perpendicular, or neither?

Parallel

Neither

Perpendicular

Perpendicular

Explanation:

Any equation of the form , such as , can be graphed by a vertical line; any equation of the form , such as , can be graphed  by a horizontal line. A vertical line and a horizontal line are perpendicular to each other.

### Example Question #1 : How To Find Out If Lines Are Perpendicular

Are the lines of the equations

and

parallel, perpendicular, or neither?

Parallel

Perpendicular

Neither

Neither

Explanation:

Write each equation in the slope-intercept form  by solving for ; the -coefficient  is the slope of the line.

Subtract  from both sides:

The line of this equation has slope .

Subtract  from both sides:

Multiply both sides by

The line of this equation has slope .

Two lines are parallel if and only if they have the same slope; this is not the case. They are perpendicular if and only if the product of their slopes is ; this is not the case, since

.

The correct response is that the lines are neither parallel nor perpendicular.

### Example Question #1 : How To Find Out If Lines Are Perpendicular

Are the lines of the equations

and

parallel, perpendicular, or neither?

Parallel

Perpendicular

Neither

Neither

Explanation:

Write each equation in the slope-intercept form  by solving for ; the -coefficient  is the slope of the line.

Subtract  from both sides:

Multiply both sides by :

The slope is the -coefficient

Multiply both sides by :

The slope is the -coefficient .

Two lines are parallel if and only if they have the same slope; this is not the case. They are perpendicular if and only if the product of their slopes is ; this is not the case, since . The lines are neither parallel nor perpendicular.

### Example Question #1 : How To Find Out If Lines Are Perpendicular

The slopes of two lines on the coordinate plane are  and 4.

True or false: the lines are perpendicular.

False

True