1. Parallel and Perpendicular Lines
Chapter 7 Section 7
Parallel and Perpendicular Lines

2. What You’ll Learn
You’ll learn to write an equation of a line that is parallel or perpendicular to the graph of a given equation and that passes through a given point.

3. Why It’s Important
Surveying Surveyors use parallel and perpendicular lines to plan construction.

4. Parallel Lines
Words: If two lines have the same slope, then they are parallel.
Model: Symbols: II
y = 2x + 2
y = 2x - 4

5. You can always check by graphing.
Example 1
You can always check by graphing.
Determine whether the graph of the equations are parallel. y = -¾x – 2 4y = -3x + 12 First, determine the slope of the lines. Write each equation in slope intercept form. The slopes are the same, so the lines are parallel.
4y = -3x + 12
y = -¾x + 3
Slope = -¾
y = -¾x -2 Slope intercept form
Slope = -¾

6. Determine whether the graph of the equations are parallel.
Your Turn
Determine whether the graph of the equations are parallel.
y = 2x
7 = 2x – y
Yes

7. Determine whether the graph of the equations are parallel.
Your Turn
Determine whether the graph of the equations are parallel.
y = -3x + 3
2y = 6x – 5 No

8. What is a parallelogram???
A parallelogram is a four-sided figure with two sets of parallel sides.

9. Determine whether figure ABCD is a parallelogram.
Example 2
Determine whether figure ABCD is a parallelogram.
Explore: To be a parallelogram,
AB and DC must be parallel.
Also, AD and BC must be parallel.
Plan: Find the slope of each segment.
D(4,5)
Solve: AB m = -1 –
1 – (-2) 3 = = -1
A(-2,2)
C(7,2)
Solve: BC m = -1 – = =
Solve: DC m = 2 – = = -1
B(1,-1)
Solve: AD m =
4 – (-2) = =

10. Example 2: Continued
AB is parallel to DC because their slopes are both -1. BC is parallel to AD because their slopes are both ½. Therefore, figure ABCD is a parallelogram.

11. If you know the slope to a line, you can use that information to write an equation of a line that is parallel to it.

12. Example 3
Write an equation in slope-intercept form of the line that is parallel to the graph y = -4x + 8 and passes through the point (1, 3). The slope of the given line is -4. So, the slope of the new line will also be -4. Find the new equation by using the point-slope form. y – y1 = m(x - x1) Point Slope Form y – 3 = -4(x - 1) Replace (x1, y1) with (1, 3) and m with -4 y – 3 = -4x + 4 Distributive Property Add 3 to both sides y = -4x + 7 An equation whose graph is parallel to the graph of 4x + y = 8 and passes through the point at (1, 3) is y = -4x + 7. Checking: Check by substituting (1, 3) into y = -4x + 7 or by graphing

13. Your Turn
Write an equation in slope-intercept form of the line that is parallel to the graph of each equation and passes through the given point.
y = 6x – 4; (2, 3)
y = 6x - 9

14. Your Turn
Write an equation in slope-intercept form of the line that is parallel to the graph of each equation and passes through the given point.
3x + 2y = 9; (2, 0)
y = -3/2x + 3

15. Perpendicular Lines
Words: If the product of the slopes of two lines is -1, then the lines are perpendicular.
Model: Below
y = -x + 3
y = x -1

16. Example 4
Determine whether the graphs of the equations are perpendicular. y = 2/3x + 1 y = -3/2x + 2 The graphs are perpendicular because the product of their slope is
y = -3/2x + 2 ∙ = -1
y = 2/3x + 1

17. Your Turn
Determine whether the graphs of the equations are perpendicular. y = 1/5x + 2 y = 5x + 1 No

18. Your Turn
Determine whether the graphs of the equations are perpendicular. y = -4x + 3 4y = x -5 Yes

19. Example 5
Write an equation in slope-intercept form of the line that is perpendicular to the graph of y = ⅓x – 2 and passes through the point (-4, 2). The slope is ⅓. A line perpendicular to the graph of y = ⅓x – 2 has slope -3. Find the new equation by using the point-slope form. y – y1 = m(x - x1) Point Slope Form y – 2 = -3(x – (-4)) Replace (x1, y1) with (-4, 2) and m with -3 y – 2 = -3x - 12 Distributive Property Add 2 to both sides y = -3x – 10 The new equation y = -3x -10. Check by substituting (-4, 2) into the equation or by graphing.

20. Your Turn
Write an equation in slope-intercept form of the line that is perpendicular to the graph of each equation and passes through the given point. y = 2x + 6; (0, 0) y = -½x

21. Your Turn
Write an equation in slope-intercept form of the line that is perpendicular to the graph of each equation and passes through the given point. 2x + 3y = 2; (3, 0) y = 3/2x – 9/2

22. Video Examples Parallel Lines 1 Parallel Lines 2 Parellal Lines 3
Perpendicular Lines 1
Perpendicular Line 2