1. Parallel and Perpendicular Lines Section 5-6

2. Goals Goal To determine whether lines are parallel, perpendicular, or neither. To write linear equations of parallel lines and perpendicular lines. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Parallel lines Perpendicular lines Opposite reciprocals

4. Definition Parallel Lines - are lines in the same plane that never intersect. Line WX is parallel to line YZ. WX || YZ.

5. These two lines are parallel. Parallel lines are lines in the same plane that have no points in common. In other words, they do not intersect. Parallel Lines As seen on the graph, parallel lines have the same slope. For y = 3x + 100, m = 3 and for y = 3x + 50, m = 3.

6. Parallel Lines

7. Identify which lines are parallel. The lines described by and both have slope. These lines are parallel. The lines described by y = x and y = x + 1 both have slope 1. These lines are parallel. Example: Identifying Parallel Lines

8. Write all equations in slope-intercept form to determine the slope. y = 2x – 3 slope-intercept form slope-intercept form Identify which lines are parallel. Example: Identifying Parallel Lines

9. Write all equations in slope-intercept form to determine the slope. 2x + 3y = 8 –2x – 2x 3y = –2x + 8 y + 1 = 3(x – 3) y + 1 = 3x – 9 – 1 y = 3x – 10 Identify which lines are parallel. Example: Continued

10. The lines described by y = 2x – 3 and y + 1 = 3(x – 3) are not parallel with any of the lines. y = 2x – 3 y + 1 = 3(x – 3) The lines described by and represent parallel lines. They each have the slope. Example: Continued

11. Equations x = 1 and y = –4 are not parallel. The lines described by y = 2x + 2 and y = 2x + 1 represent parallel lines. They each have slope 2. y = 2x + 1 y = 2x + 2 y = –4 x = 1 Identify which lines are parallel. y = 2x + 2; y = 2x + 1; y = –4; x = 1 Your Turn:

12. Identify which lines are parallel. Write all equations in slope-intercept form to determine the slope. Slope-intercept form y = 3x Slope-intercept form Your Turn:

13. Identify which lines are parallel. Write all equations in slope-intercept form to determine the slope. –3x + 4y = 32 +3x 4y = 3x + 32 y – 1 = 3(x + 2) y – 1 = 3x + 6 + 1 y = 3x + 7 Your Turn: Continued

14. The lines described by –3x + 4y = 32 and y = + 8 have the same slope, but they are not parallel lines. They are the same line. –3x + 4y = 32 y – 1 = 3(x + 2) y = 3x The lines described by y = 3x and y – 1 = 3(x + 2) represent parallel lines. They each have slope 3. Your Turn: Continued

15. Definition Perpendicular Lines – lines that intersect to form 90 o angles, or right angles. Line RS is perpendicular to line TU. RS | TU. __

16. Perpendicular Lines Perpendicular lines have slopes that are opposite reciprocals.

17. Definition Opposite Reciprocals – two numbers whose product is – 1. –Example: To find the opposite reciprocal (also called negative reciprocal) of – 3/4, first find the reciprocal, –4/3. Then write its opposite, 4/3. Since -3/4 ∙ 4/3 = -1, therefore 4/3 is the opposite reciprocal of -3/4.

18. Helpful Hint If you know the slope of a line, the slope of a perpendicular line will be the "opposite reciprocal.” Opposite Reciprocals

19. Identify which lines are perpendicular: y = 3; x = –2; y = 3x;. The graph described by y = 3 is a horizontal line, and the graph described by x = –2 is a vertical line. These lines are perpendicular. y = 3 x = –2 y =3x The slope of the line described by y = 3x is 3. The slope of the line described by is. Example: Indentifying Perpendicular Lines

20. Identify which lines are perpendicular: y = 3; x = –2; y = 3x;. y = 3 x = –2 y =3x These lines are perpendicular because the product of their slopes is –1. Example: Continued

21. Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4); x = 3; y = The graph described by x = 3 is a vertical line, and the graph described by y = –4 is a horizontal line. These lines are perpendicular. The slope of the line described by y – 6 = 5(x + 4) is 5. The slope of the line described by y = is y = –4 x = 3 y – 6 = 5(x + 4) Your Turn:

22. Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4); x = 3; y = These lines are perpendicular because the product of their slopes is –1. y = –4 x = 3 y – 6 = 5(x + 4) Your Turn: Continued

23. Write an equation in slope-intercept form for the line that passes through (4, 10) and is parallel to the line described by y = 3x + 8. Step 1 Find the slope of the line. y = 3x + 8 The slope is 3. The parallel line also has a slope of 3. Step 2 Write the equation in point-slope form. Use the point-slope form. y – y 1 = m(x – x 1 ) y – 10 = 3(x – 4) Substitute 3 for m, 4 for x 1, and 10 for y 1. Example: Writing Equations of Parallel and Perpendicular Lines

24. Step 3 Write the equation in slope-intercept form. y – 10 = 3(x – 4) y – 10 = 3x – 12) y = 3x – 2 Distribute 3 on the right side. Add 10 to both sides. Example: Continued

25. Write an equation in slope-intercept form for the line that passes through (2, –1) and is perpendicular to the line described by y = 2x – 5. Step 1 Find the slope of the line. y = 2x – 5 The slope is 2. The perpendicular line has a slope of because Step 2 Write the equation in point-slope form. Use the point-slope form. y – y 1 = m(x – x 1 ) Substitute for m, –1 for y 1, and 2 for x 1. Example: Writing Equations of Parallel and Perpendicular Lines

26. Step 3 Write the equation in slope-intercept form. Distribute on the right side. Subtract 1 from both sides. Example: Continued

27. Write an equation in slope-intercept form for the line that passes through (5, 7) and is parallel to the line described by y = x – 6. Step 1 Find the slope of the line. Step 2 Write the equation in point-slope form. Use the point-slope form. y = x –6 The slope is. The parallel line also has a slope of. y – y 1 = m(x – x 1 ) Your Turn:

28. Step 3 Write the equation in slope-intercept form. Add 7 to both sides. Distribute on the right side. Your Turn: Continued

29. Write an equation in slope-intercept form for the line that passes through (–5, 3) and is perpendicular to the line described by y = 5x. Step 1 Find the slope of the line. y = 5x The slope is 5. Step 2 Write the equation in point-slope form. Use the point-slope form. The perpendicular line has a slope of because. y – y 1 = m(x – x 1 ) Your Turn:

30. Step 3 Write in slope-intercept form. Add 3 to both sides. Distribute on the right side. Write an equation in slope-intercept form for the line that passes through (–5, 3) and is perpendicular to the line described by y = 5x. Your Turn: Continued

31. Joke Time What do clouds wear under their shorts? Thunderpants! What kind of music do mummies listen to? Wrap! Why is there no gambling in Africa? Too many Cheetahs!

32. Assignment 5-6 Exercises Pg. 361 - 362: #6 – 40 even