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
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
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
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)
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?
47. Homework
p. 336 #1-39 odd
“It is perhaps a more fortunate destiny to have a taste for collecting
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