Find the equation of the line parallel to that passes through the point . Then, find the equation of the line perpendicular to that passes through the point . How are the two lines that you found related?

### Slope of Parallel and Perpendicular Lines

Consider two lines. There are three ways that the two lines can interact:

- They are parallel and so they never intersect.
- They are perpendicular and so they intersect at a right angle.
- They intersect, but they are not perpendicular.

Recall that the **slope** of a line is a measure of its **steepness**. For a line written in the form , “” is the slope. Given two lines, their slopes can help you to determine whether the lines are parallel, perpendicular, or neither.

In the past you learned that **two lines are parallel if and only if they have the same slope**.

In the past you also learned that **two lines are perpendicular if and only if they have slopes that are opposite reciprocals**. This means that if the slope of one line is , the slope of a line perpendicular to it will be . Another way of thinking about this is that the product of the slopes of perpendicular lines will always be -1. (Note that ).

#### Determining the Distinction Between Two Lines

Consider two lines and with . Note that these two lines have the same slope, .

How do you know that the two lines are distinct (not the same line?)

The two lines are distinct because they have different -intercepts. The first line has a -intercept at and the second line has a -intercept at .

Use algebra to find the point of intersection of the lines. What happens?

You can use substitution to attempt to find the point of intersection.

and

Therefore:

This is a contradiction because it was stated that . Therefore, these two lines do not have a point of intersection. This means the lines must be parallel. *This proves that if two lines have the same slope, then they are parallel.*

Consider rectangle with , and perpendicular lines and .

#### Determining Similarity

Find the length of . Then, show that is similar to .

Because it is a rectangle, . The two triangles are similar because they have congruent angles. Let and label all angles in the picture in terms of .

You can see that each of the three triangles in the picture have the same angle measures, so they must all be similar. In particular, is similar to .

Now, Use the fact that is similar to to find the length of . Then, find the slopes of lines and and show that their product is -1.

Because is similar to , the following proportion is true:

Solving this proportion you have that

The slopes of the lines can be found using . The slope of line is and the slope of is .

The product of the slopes is .

*This proves that if two lines are perpendicular, then their slopes will be opposite reciprocals (the product of the slopes will be -1).*

### Examples

#### Example 1

Earlier, you were asked how two lines that you found are related.

To find the equation of the line parallel to that passes through the point , remember that parallel lines must have equal slopes. This means that the new line must have a slope of and pass through the point . All you need to do is solve for the -intercept.

The equation of the line is .

To find the equation of the line perpendicular to that passes through the point , remember that perpendicular lines will have opposite reciprocal slopes. This means that the new line must have a slope of and pass through the point . Again, all you need to do is solve for the -intercept.

The equation of the line is

The two lines that were found and are also perpendicular. Note that they have opposite reciprocal slopes.

#### Example 2

Consider two parallel lines and with . Show that .

Suppose . You can solve a system of equations to find the point of intersection of the two lines.

and

Therefore:

If , then this point exists so the lines intersect. This is a contradiction because it was stated that the lines were parallel. Therefore, must be equal to . *This proves that if two lines are parallel then they must have the same slope. *

#### Example 3

Consider two lines intersecting at the origin as shown below. Find the lengths of the legs of each triangle. Then, show that is similar to .

The lengths of the legs of the triangles are shown below.

with a ratio of by . and Also .

#### Example 4

Using the picture from #3, find the slopes of lines and and verify that their product is -1. Then use the fact that is similar to to show that and must be perpendicular.

The slope of line is and the slope of is . The product of the slopes is .

Because is similar to , their corresponding angles must be congruent. This means that:

Also, because they are right triangles:

By substitution, . Because and form a straight line, the sum of their measures must be . Therefore, must be .

Because , and must be perpendicular. *This proves that if two lines have opposite reciprocal slopes, then they are perpendicular.*

### Review

1. Describe the three ways that two lines could interact. Draw a picture of each.

2. What does it mean for two lines to be parallel? How are the slopes of parallel lines related?

3. What does it mean for two lines to be perpendicular? How are the slopes of perpendicular lines related?

4. Use algebra to show why the lines and (lines with the same slope) must be parallel.

5. Use the method from Example B and Example C to show why the slopes of lines and must be opposite reciprocals. *Assume that* *is a rectangle. *

6. Find the line parallel to that passes through

7. Find the line perpendicular to that passes through

8. Find the line parallel to that passes through

9. Find the line perpendicular to that passes through

10. Find the line parallel to that passes through

11. Find the line perpendicular to that passes through

12. Find the line parallel to that passes through

13. Find the line perpendicular to that passes through

14. Line passes through the point and . Line passes through the points and . Are lines and parallel, perpendicular, or neither?

15. Line passes through the point and . Line passes through the points and . Are lines and parallel, perpendicular, or neither?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.4.