### All ACT Math Resources

## Example Questions

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

Which of the following lines is perpendicular to the line ?

**Possible Answers:**

**Correct answer:**

Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.

The slope of this line is . The negative reciprocal will be , which will be the slope of the perpendicular line.

Now we need to find the answer choice with this slope by converting to slope-intercept form.

This equation has a slope of , and must be our answer.

### Example Question #1 : Perpendicular Lines

Which of the following lines is perpindicular to

**Possible Answers:**

None of the other answers

**Correct answer:**

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, , where m is the slope of the line. Two lines are perpindicular to each other if one slope is the negative and reciprocal of the other.

The first step of this problem is to get it into the form, , which is . Now we know that the slope, m, is . The reciprocal of that is , and the negative of that is . Therefore, any line that has a slope of will be perpindicular to the original line.

### Example Question #23 : Perpendicular Lines

Which of the following lines is perpendicular to the line with the given equation:

?

**Possible Answers:**

**Correct answer:**

First we must recognize that the equation is given in slope-intercept form, where is the slope of the line.

Two lines are perpendicular if and only if the product of their slopes is . In other words, the slope of a line that is perpendicular to a given line is the negative reciprocal of that slope.

Thus, for a line with a given slope of 3, the line perpendicular to that slope must be the negative reciprocal of 3, or .

To double check that that does indeed give a product of when multiplied by three simply compute the product:

### Example Question #24 : Perpendicular Lines

Are the following two lines parallel, perpendicular, or neither:

and

**Possible Answers:**

They are both perpendicular and parallel.

They are neither perpendicular or parallel.

More information is needed.

They are perpendicular.

They are parallel.

**Correct answer:**

They are perpendicular.

Perpendicular lines have slopes whose product is .

The slope is controlled by the coefficient, from the genral form of the slope-intercept equation:

Thus the two lines are perpendicular because:

has

and

has

which when multiplied together results in,

.

### Example Question #25 : Perpendicular Lines

Are the following two lines perpendicular:

**Possible Answers:**

**Correct answer:**

For two lines to be perpendicular, their slopes have to have a product of . Find the slopes by the coefficient in front of the .

and so the two lines are perpendicular. The y-intercept does not matter for determine perpendicularity.

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

Are the lines described by the equations and perpendicular to one another? Why or why not?

**Possible Answers:**

No, because the product of their slopes is .

Yes, because the product of their slopes is .

No, because the product of their slopes is not .

Yes, because the product of their slopes is not .

**Correct answer:**

No, because the product of their slopes is not .

If the slopes of two lines can be calculated, an easy way to determine whether they are perpendicular is to multiply their slopes. If the product of the slopes is , then the lines are perpendicular.

In this case, the slope of the line is and the slope of the line is .

Since , the slopes are not perpendicular.

### Example Question #31 : Perpendicular Lines

Line , which follows the equation , intersects line at . If line also passes through , are and perpendicular?

**Possible Answers:**

No, because the product of their slopes is

Yes, because the product of their slopes is not .

Yes, because the product of their slopes is .

No, because the product of their slope is not .

**Correct answer:**

Yes, because the product of their slopes is .

The product of perpendicular slopes is always . Knowing this, and seeing that the slope of line is , we know any perpendicular line will have a slope of .

Since line passes through and , we can use the slope equation:

Since the two slopes' product is , the lines are perpendicular.

### Example Question #32 : Perpendicular Lines

Are the following two lines perpendicular:

and

**Possible Answers:**

**Correct answer:**

For two lines to be perpendicular they have to have slopes that multiply to get . The slope is found from the in the general equation: .

For the first line, and for the second . and so the lines are not perpendicular.

### Example Question #2 : Perpendicular Lines

Which of the following equations represents a line that is perpendicular to the line with points and ?

**Possible Answers:**

**Correct answer:**

If lines are perpendicular, then their slopes will be negative reciprocals.

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

Because we know that our given line's slope is , the slope of the line perpendicular to it must be .

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