Parallel and Perpendicular Lines

1. Chapter 8
Systems of Equations and Inequalities

2. Section 8-1
Parallel and Perpendicular Lines

3. Essential Questions
✤ How do you determine if two lines are parallel or perpendicular?
✤ How do you write equations of parallel and perpendicular lines?
✤ Where you’ll see this:
✤ Sports, travel, safety

4. Vocabulary
1. Negative Reciprocals:

5. Vocabulary
1. Negative Reciprocals: Two rational numbers whose product is -1

6. Vocabulary
1. Negative Reciprocals: Two rational numbers whose product is -1
a b
and − are negative reciprocals
b a

7. Vocabulary
1. Negative Reciprocals: Two rational numbers whose product is -1
a b
and − are negative reciprocals
b a
2 3
and − are negative reciprocals
3 2

8. Parallel Lines

9. Parallel Lines
✤ Don’t intersect

10. Parallel Lines
✤ Don’t intersect
✤ Same slope

11. Parallel Lines
✤ Don’t intersect
✤ Same slope

12. Parallel Lines
✤ Don’t intersect
✤ Same slope
✤ If two lines are parallel, then they have the same slope

13. Parallel Lines
✤ Don’t intersect
✤ Same slope
✤ If two lines are parallel, then they have the same slope
✤ If two lines have the same slope, then they are parallel

14. Perpendicular Lines

15. Perpendicular Lines
✤ Intersect at 90 degree angles

16. Perpendicular Lines
✤ Intersect at 90 degree angles
✤ Slopes are negative reciprocals

17. Perpendicular Lines
✤ Intersect at 90 degree angles
✤ Slopes are negative reciprocals

18. Perpendicular Lines
✤ Intersect at 90 degree angles
✤ Slopes are negative reciprocals
✤ If two lines are perpendicular, then the product of their slopes is -1

19. Perpendicular Lines
✤ Intersect at 90 degree angles
✤ Slopes are negative reciprocals
✤ If two lines are perpendicular, then the product of their slopes is -1
✤ If the product of the slopes of two lines is -1, then they are
perpendicular

20. Example 1
Suppose you have a line that has a slope of 4. What would be the slope
of:
a. the perpendicular line?
b. the parallel line?

21. Example 1
Suppose you have a line that has a slope of 4. What would be the slope
of:
a. the perpendicular line?
1
m=−
4
b. the parallel line?

22. Example 1
Suppose you have a line that has a slope of 4. What would be the slope
of:
a. the perpendicular line?
1
m=−
4
b. the parallel line?
m=4

23. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.

24. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12

25. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x

26. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12

27. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12
3 3

28. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12
3 3
y = 2x + 4

29. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12
3 3
y = 2x + 4
m=2

30. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12
+6x +6x
3y = 6x + 12
3 3
y = 2x + 4
m=2
(-1, 3)

31. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12 y − y1 = m(x − x1 )
+6x +6x
3y = 6x + 12
3 3
y = 2x + 4
m=2
(-1, 3)

32. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12 y − y1 = m(x − x1 )
+6x +6x
y − 3 = 2(x + 1)
3y = 6x + 12
3 3
y = 2x + 4
m=2
(-1, 3)

33. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12 y − y1 = m(x − x1 )
+6x +6x
y − 3 = 2(x + 1)
3y = 6x + 12
3 3 y − 3 = 2x + 2
y = 2x + 4
m=2
(-1, 3)

34. Example 2
Write the equation for the line that passes through (-1, 3) and is parallel
to the line 3y - 6x = 12.
3y − 6x = 12 y − y1 = m(x − x1 )
+6x +6x
y − 3 = 2(x + 1)
3y = 6x + 12
3 3 y − 3 = 2x + 2
y = 2x + 4 y = 2x + 5
m=2
(-1, 3)

35. Formulas
Slope-intercept:
Point-slope:

36. Formulas
Slope-intercept:
y = mx + b
Point-slope:

37. Formulas
Slope-intercept:
y = mx + b
Point-slope:
y − y1 = m(x − x1 )

38. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4

39. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
m=
3

40. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
m= (2, 7)
3

41. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
m= (2, 7)
3
y − y1 = m(x − x1 )

42. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
y − 7 = (x − 2)
4 3
m= (2, 7)
3
y − y1 = m(x − x1 )

43. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
y − 7 = (x − 2)
4 3
m= (2, 7)
3 4 8
y−7= x−
y − y1 = m(x − x1 ) 3 3

44. Example 3
Write the equation for the line that passes through (2, 7) and is
perpendicular to
3
y=− x+6
4
4
y − 7 = (x − 2)
4 3
m= (2, 7)
3 4 8
y−7= x−
y − y1 = m(x − x1 ) 3 3
4 13
y= x+
3 3

45. Quick Questions
1. What is true about the slopes of all horizontal lines?
2. If you knew the coordinates of four vertices of a quadrilateral, how
could you use slope to determine if the figure is a parallelogram?

46. Homework

47. Homework
p. 336 #1-39 odd
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