### All ACT Math Resources

## Example Questions

### Example Question #31 : Perpendicular Lines

What is the slope of any line perpendicular to 2*y* = 4*x* +3 ?

**Possible Answers:**

– ½

½

– 4

2

**Correct answer:**

– ½

First, we must solve the equation for *y* to determine the slope: *y* = 2*x* + ^{3}/_{2}

By looking at the coefficient in front of *x*, we know that the slope of this line has a value of 2. To fine the slope of any line perpendicular to this one, we take the negative reciprocal of it:

slope = *m* , perpendicular slope = – ^{1}/_{m}

slope = 2 , perpendicular slope = – ^{1}/_{2}

### Example Question #1 : How To Find The Slope Of A Perpendicular Line

What line is perpendicular to 2x + y = 3 at (1,1)?

**Possible Answers:**

x – 2y = -1

x + 2y - 3

3x + 2y = 1

2x + 3y = 1

**Correct answer:**

x – 2y = -1

Find the slope of the given line. The perpendicular slope will be the opposite reciprocal of the original slope. Use the slope-intercept form (y = mx + b) and substitute in the given point and the new slope to find the intercept, b. Convert back to standard form of an equation: ax + by = c.

### Example Question #1 : How To Find The Slope Of A Perpendicular Line

What is the slope of the line perpendicular to the line given by the equation

*6x – 9y +14 = 0*

**Possible Answers:**

-2/3

-1/6

2/3

6

-3/2

**Correct answer:**

-3/2

First rearrange the equation so that it is in slope-intercept form, resulting in *y=2/3 x + 14/9*. The slope of this line is 2/3, so the slope of the line perpendicular will have the opposite reciprocal as a slope, which is -3/2.

### Example Question #1 : How To Find The Slope Of A Perpendicular Line

What is the slope of the line perpendicular to the line represented by the equation y = -2x+3?

**Possible Answers:**

2/3

-2/3

1/2

-1/2

2

**Correct answer:**

1/2

Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2.

### Example Question #1 : How To Find The Slope Of A Perpendicular Line

Find the slope of a line perpendicular to the line y = –3x – 4.

**Possible Answers:**

1/3

–3

4

1/4

**Correct answer:**

1/3

First we must find the slope of the given line. The slope of y = –3x – 4 is –3. The slope of the perpendicular line is the negative reciprocal. This means you change the sign of the slope to its opposite: in this case to 3. Then find the reciprocal by switching the denominator and numerator to get 1/3; therefore the slope of the perpendicular line is 1/3.

### Example Question #1 : How To Find The Slope Of A Perpendicular Line

What is the slope of a line perpendicular to the following:

**Possible Answers:**

**Correct answer:**

The question puts the line in point-slope form y *– *y_{1} = m(x *–* x_{1}), where m is the slope. Therefore, the slope of the original line is 1/2. A line perpendicular to another has a slope that is the negative reciprocal of the slope of the other line. The negative reciprocal of the original line is *–*2, and is thus the slope of its perpendicular line.

### Example Question #1 : How To Find The Slope Of A Perpendicular Line

A line is defined by the following equation:

What is the slope of a line that is perpendicular to the line above?

**Possible Answers:**

**Correct answer:**

The equation of a line is where is the slope.

Rearrange the equation to match this:

For the perpendicular line, the slope is the negative reciprocal;

therefore

### Example Question #1 : How To Find The Slope Of A Perpendicular Line

Which of the following lines is perpendicular to the line passing through the points ?

**Possible Answers:**

**Correct answer:**

First, you must find the slope of the line given to you. Remember that the slope is calculated:

Thus, for our data, this is:

Now, the perpendicular slope to this is *opposite* and *reciprocal*. Hence, it must be . This only holds for the equation

To know this, solve the equation for the format . This will let you find the slope very quickly, for it is . First, add to both sides:

Next, divide everything by :

You really just need to pay attention to the term. This reduces to , which is just what you need!

### Example Question #2 : How To Find The Slope Of A Perpendicular Line

What is the slope of a line that is perpendicular to the equation given by:

**Possible Answers:**

**Correct answer:**

Perependicular lines have slopes whose product is .

and so the answer is

### Example Question #3 : How To Find The Slope Of A Perpendicular Line

What is the slope of a line perpendicular to line , which runs through and ?

**Possible Answers:**

**Correct answer:**

To find the slope of a perpendicular line, we take the reciprocal of the known slope , where .

The easy way to do this is to simply take the fraction (a whole slope can be made into a fraction by placing in the denominator), exchange the numerator and denominator, then multiply the fraction by

However, if we attempt to follow this procedure, we get:

, which is undefined.

Thus, our perpendicular line (which is a vertical line) has an undefined slope.

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