<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

Parallel and Perpendicular Lines in the Coordinate Plane

Lines with slopes that are equal or opposite reciprocals of each other.

Estimated7 minsto complete
%
Progress
Practice Parallel and Perpendicular Lines in the Coordinate Plane

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated7 minsto complete
%
Slope of Parallel and Perpendicular Lines

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:

1. They are parallel and so they never intersect.
2. They are perpendicular and so they intersect at a right angle.
3. 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?

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

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

Vocabulary Language: English

Parallel

Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.

Perpendicular

Perpendicular lines are lines that intersect at a $90^{\circ}$ angle. The product of the slopes of two perpendicular lines is -1.

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$