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# Geometry Section 3-3 1112

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Slopes of Lines

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### Geometry Section 3-3 1112

1. 1. Section 3-3 Slopes of LinesMonday, December 19, 2011
2. 2. Essential Questions How do you ﬁnd slopes of lines? How do you use slope to identify parallel and perpendicular lines?Monday, December 19, 2011
3. 3. Vocabulary 1. Slope: 2. Rate of Change:Monday, December 19, 2011
4. 4. Vocabulary 1. Slope: The ratio of the vertical change to the horizontal change between two points 2. Rate of Change:Monday, December 19, 2011
5. 5. Vocabulary 1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x 2. Rate of Change:Monday, December 19, 2011
6. 6. Vocabulary 1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x rise over run 2. Rate of Change:Monday, December 19, 2011
7. 7. Vocabulary 1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x rise over run 2. Rate of Change: A way to describe slopeMonday, December 19, 2011
8. 8. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between the horizontal lines. Distance between vertical Distance between horizontalMonday, December 19, 2011
9. 9. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical Distance between horizontal xMonday, December 19, 2011
10. 10. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical Distance between horizontal x AMonday, December 19, 2011
11. 11. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
12. 12. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
13. 13. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
14. 14. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
15. 15. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
16. 16. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical 5 Units B Distance between horizontal x AMonday, December 19, 2011
17. 17. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical 5 Units B Distance between horizontal x 7 Units AMonday, December 19, 2011
18. 18. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical 5 Units B Distance between horizontal x 7 Units A Up 7, Right 5Monday, December 19, 2011
19. 19. Slope FormulaMonday, December 19, 2011
20. 20. Slope Formula y 2 − y1 m= x 2 − x1 for points (x 1,y 1),(x 2 ,y 2 )Monday, December 19, 2011
21. 21. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1)Monday, December 19, 2011
22. 22. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1)Monday, December 19, 2011
23. 23. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 )Monday, December 19, 2011
24. 24. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 ) y 2 − y1 m= x 2 − x1Monday, December 19, 2011
25. 25. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 ) y 2 − y1 1− 4 m= = x 2 − x 1 8 − (−3)Monday, December 19, 2011
26. 26. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 ) y 2 − y1 1− 4 −3 m= = = x 2 − x 1 8 − (−3) 11Monday, December 19, 2011
27. 27. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 ) y 2 − y1 1− 4 −3 m= = = x 2 − x 1 8 − (−3) 11 Down 3, Right 11Monday, December 19, 2011
28. 28. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7)Monday, December 19, 2011
29. 29. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y1 m= x 2 − x1Monday, December 19, 2011
30. 30. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y 1 7 − (−1) m= = x 2 − x 1 −3 − 5Monday, December 19, 2011
31. 31. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y 1 7 − (−1) 8 m= = = x 2 − x 1 −3 − 5 −8Monday, December 19, 2011
32. 32. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y 1 7 − (−1) 8 m= = = = −1 x 2 − x 1 −3 − 5 −8Monday, December 19, 2011
33. 33. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y 1 7 − (−1) 8 m= = = = −1 x 2 − x 1 −3 − 5 −8 Down 1, Right 1Monday, December 19, 2011
34. 34. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7)Monday, December 19, 2011
35. 35. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 m= x 2 − x1Monday, December 19, 2011
36. 36. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 m= = x 2 − x 1 −1− (−1)Monday, December 19, 2011
37. 37. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 5 m= = = x 2 − x 1 −1− (−1) 0Monday, December 19, 2011
38. 38. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 5 m= = = x 2 − x 1 −1− (−1) 0 UndeﬁnedMonday, December 19, 2011
39. 39. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 5 m= = = x 2 − x 1 −1− (−1) 0 Undeﬁned Up 5, Right 0Monday, December 19, 2011
40. 40. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 5 m= = = x 2 − x 1 −1− (−1) 0 Undeﬁned Up 5, Right 0 Vertical LineMonday, December 19, 2011
41. 41. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4)Monday, December 19, 2011
42. 42. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 m= x 2 − x1Monday, December 19, 2011
43. 43. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 m= = x 2 − x 1 −2 − 3Monday, December 19, 2011
44. 44. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 0 m= = = x 2 − x 1 −2 − 3 −5Monday, December 19, 2011
45. 45. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 0 m= = = x 2 − x 1 −2 − 3 −5 Up 0, Left 5Monday, December 19, 2011
46. 46. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 0 m= = = x 2 − x 1 −2 − 3 −5 Up 0, Left 5 Horizontal LineMonday, December 19, 2011
47. 47. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 0 m= = = =0 x 2 − x 1 −2 − 3 −5 Up 0, Left 5 Horizontal LineMonday, December 19, 2011
48. 48. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was \$48.9 million. In 2005, the total sales were \$85.9 million. If sales increase at the same rate, what will the total sales be in 2015?Monday, December 19, 2011
49. 49. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was \$48.9 million. In 2005, the total sales were \$85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 =Monday, December 19, 2011
50. 50. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was \$48.9 million. In 2005, the total sales were \$85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Monday, December 19, 2011
51. 51. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was \$48.9 million. In 2005, the total sales were \$85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Every 5 years, sales increase by \$37 millionMonday, December 19, 2011
52. 52. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was \$48.9 million. In 2005, the total sales were \$85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Every 5 years, sales increase by \$37 million 85.9 + 2(37) =Monday, December 19, 2011
53. 53. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was \$48.9 million. In 2005, the total sales were \$85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Every 5 years, sales increase by \$37 million 85.9 + 2(37) = 159.9Monday, December 19, 2011
54. 54. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was \$48.9 million. In 2005, the total sales were \$85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Every 5 years, sales increase by \$37 million 85.9 + 2(37) = 159.9 In 2015, sales should be about \$159.9 millionMonday, December 19, 2011
55. 55. Postulates Slopes of parallel lines: Slopes of perpendicular lines:Monday, December 19, 2011
56. 56. Postulates Slopes of parallel lines: Two lines will be parallel IFF they have the same slope. All vertical lines are parallel. Slopes of perpendicular lines:Monday, December 19, 2011
57. 57. Postulates Slopes of parallel lines: Two lines will be parallel IFF they have the same slope. All vertical lines are parallel. Slopes of perpendicular lines: Two lines will be perpendicular IFF the product of their slopes is −1. Vertical and horizontal lines are perpendicular.Monday, December 19, 2011
58. 58. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.Monday, December 19, 2011
59. 59. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y m(FG ) = 2 1 x 2 − x1Monday, December 19, 2011
60. 60. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) m(FG ) = 2 1 = x 2 − x1 −2 − 1Monday, December 19, 2011
61. 61. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3Monday, December 19, 2011
62. 62. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y m(HJ ) = 2 1 x 2 − x1Monday, December 19, 2011
63. 63. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y 3−0 m(HJ ) = 2 1 = x 2 − x1 6 − 5Monday, December 19, 2011
64. 64. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y 3−0 3 m(HJ ) = 2 1 = = x 2 − x1 6 − 5 1Monday, December 19, 2011
65. 65. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y 3−0 3 m(HJ ) = 2 1 = = =3 x 2 − x1 6 − 5 1Monday, December 19, 2011
66. 66. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y 3−0 3 m(HJ ) = 2 1 = = =3 x 2 − x1 6 − 5 1 NeitherMonday, December 19, 2011
67. 67. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y xMonday, December 19, 2011
68. 68. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 m= x 2 − x1 xMonday, December 19, 2011
69. 69. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 m= = x 2 − x 1 −2 − 2 xMonday, December 19, 2011
70. 70. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 xMonday, December 19, 2011
71. 71. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 m=− 4 xMonday, December 19, 2011
72. 72. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
73. 73. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
74. 74. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
75. 75. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
76. 76. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
77. 77. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 M 3 Q m=− N 4 xMonday, December 19, 2011
78. 78. Check Your Understanding Use problems 1-11 to check the ideas from this lessonMonday, December 19, 2011
79. 79. Problem SetMonday, December 19, 2011
80. 80. Problem Set p. 191 #13-39 odd “I have found power in the mysteries of thought.” - EuripidesMonday, December 19, 2011