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# Int Math 2 Section 6-1

Distance in the Coordinate Plane

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### Int Math 2 Section 6-1

1. 1. Chapter 6Graphing Functions
2. 2. Section 6-1Distance in the Coordinate Plane
3. 3. Essential QuestionsHow do you use the distance formula to find the distancebetween two points?How do you use the midpoint formula?Where you’ll see this: Geography, market research, community service, architecture
4. 4. Vocabulary1. Coordinate Plane:2. Quadrants:3. x-axis:4. y-axis:5. Ordered Pairs:6. Origin:
5. 5. Vocabulary1. Coordinate Plane: Two number lines drawn perpendicular to each other2. Quadrants:3. x-axis:4. y-axis:5. Ordered Pairs:6. Origin:
6. 6. Vocabulary1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points2. Quadrants:3. x-axis:4. y-axis:5. Ordered Pairs:6. Origin:
7. 7. Vocabulary1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points2. Quadrants: Four areas created by the coordinate plane3. x-axis:4. y-axis:5. Ordered Pairs:6. Origin:
8. 8. Vocabulary1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points2. Quadrants: Four areas created by the coordinate plane3. x-axis: The horizontal axis on the coordinate plane4. y-axis:5. Ordered Pairs:6. Origin:
9. 9. Vocabulary1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points2. Quadrants: Four areas created by the coordinate plane3. x-axis: The horizontal axis on the coordinate plane4. y-axis: The vertical axis on the coordinate plane5. Ordered Pairs:6. Origin:
10. 10. Vocabulary1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points2. Quadrants: Four areas created by the coordinate plane3. x-axis: The horizontal axis on the coordinate plane4. y-axis: The vertical axis on the coordinate plane5. Ordered Pairs: Give us points in the form (x, y)6. Origin:
11. 11. Vocabulary1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points2. Quadrants: Four areas created by the coordinate plane3. x-axis: The horizontal axis on the coordinate plane4. y-axis: The vertical axis on the coordinate plane5. Ordered Pairs: Give us points in the form (x, y)6. Origin: The point (0, 0), which is where the x-axis and y-axis meet
12. 12. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD.
13. 13. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. A
14. 14. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. A B
15. 15. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C A B
16. 16. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. D C A B
17. 17. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. D C A B
18. 18. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. D C AB = 4 −(−2) A B
19. 19. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. D C AB = 4 −(−2) = 4 + 2 A B
20. 20. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. D C AB = 4 −(−2) = 4 + 2 = 6 A B
21. 21. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C D AB = 4 −(−2) = 4 + 2 = 6 = 6 A B
22. 22. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C D AB = 4 −(−2) = 4 + 2 = 6 = 6 AD = 5−(−2) A B
23. 23. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C D AB = 4 −(−2) = 4 + 2 = 6 = 6 AD = 5−(−2) = 5+ 2 A B
24. 24. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C D AB = 4 −(−2) = 4 + 2 = 6 = 6 AD = 5−(−2) = 5+ 2 = 7 A B
25. 25. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C D AB = 4 −(−2) = 4 + 2 = 6 = 6 AD = 5−(−2) = 5+ 2 = 7 = 7 A B
26. 26. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C D AB = 4 −(−2) = 4 + 2 = 6 = 6 AD = 5−(−2) = 5+ 2 = 7 = 7 A B Area = lw
27. 27. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C D AB = 4 −(−2) = 4 + 2 = 6 = 6 AD = 5−(−2) = 5+ 2 = 7 = 7 A B Area = lw = 6(7)
28. 28. Example 1 The vertices of rectangle ABCD are as follows: A = (-2, -2),B = (4, -2), C = (4, 5), and D = (-2, 5). Find the area of ABCD. C D AB = 4 −(−2) = 4 + 2 = 6 = 6 AD = 5−(−2) = 5+ 2 = 7 = 7 A B Area = lw = 6(7) = 42 square units
29. 29. Example 2Find the distance between the points (0, 4) and (4, 0).
30. 30. Example 2Find the distance between the points (0, 4) and (4, 0).
31. 31. Example 2Find the distance between the points (0, 4) and (4, 0).
32. 32. Example 2Find the distance between the points (0, 4) and (4, 0).
33. 33. Example 2Find the distance between the points (0, 4) and (4, 0).
34. 34. Example 2Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c
35. 35. Example 2Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c 2 2 2 0−4 + 4−0 = c
36. 36. Example 2Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c 2 2 2 0−4 + 4−0 = c 2 2 2 4 +4 = c
37. 37. Example 2Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c 2 2 2 0−4 + 4−0 = c 2 2 2 4 +4 = c 2 16+16 = c
38. 38. Example 2Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c 2 2 2 0−4 + 4−0 = c 2 2 2 4 +4 = c 2 16+16 = c 2 32 = c
39. 39. Example 2Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c 2 2 2 0−4 + 4−0 = c 2 2 2 4 +4 = c 2 16+16 = c 2 32 = c 2 c = ± 32
40. 40. Example 2Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c 2 2 2 0−4 + 4−0 = c 2 2 2 4 +4 = c 2 16+16 = c 2 32 = c 2 c = ± 32 c = 32
41. 41. Example 2Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c 2 2 2 0−4 + 4−0 = c 2 2 2 4 +4 = c 2 16+16 = c 2 32 = c 2 c = ± 32 c = 32 units
42. 42. Distance Formula:Midpoint Formula:
43. 43. Distance Formula: d = (x1 − x2 )2 +( y1 − y2 )2 , for points (x1 , y1 ),(x2 , y2 )Midpoint Formula:
44. 44. Distance Formula: d = (x1 − x2 )2 +( y1 − y2 )2 , for points (x1 , y1 ),(x2 , y2 ) This is nothing more than the Pythagorean Formula solved for c.Midpoint Formula:
45. 45. Distance Formula: d = (x1 − x2 )2 +( y1 − y2 )2 , for points (x1 , y1 ),(x2 , y2 ) This is nothing more than the Pythagorean Formula solved for c.  x1 + x2 y1 + y2 Midpoint Formula: M =  ,  , for points (x1 , y1 ),(x2 , y2 )  2 2 
46. 46. Distance Formula: d = (x1 − x2 )2 +( y1 − y2 )2 , for points (x1 , y1 ),(x2 , y2 ) This is nothing more than the Pythagorean Formula solved for c.  x1 + x2 y1 + y2 Midpoint Formula: M =  ,  , for points (x1 , y1 ),(x2 , y2 )  2 2  This is nothing more than averaging the x and y coordinates.
47. 47. Example 3The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). a. What kind of quadrilateral does ABCD appear to be?
48. 48. Example 3The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). a. What kind of quadrilateral does ABCD appear to be? A
49. 49. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). a. What kind of quadrilateral does ABCD appear to be? AB
50. 50. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). a. What kind of quadrilateral does ABCD appear to be? AB C
51. 51. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). a. What kind of quadrilateral does ABCD appear to be? AB D C
52. 52. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). a. What kind of quadrilateral does ABCD appear to be? AB D C
53. 53. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). a. What kind of quadrilateral does ABCD appear to be? A This quadrilateral appearsB to be a parallelogram D C
54. 54. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. AB D C
55. 55. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 AB = (2 −(−5)) +(4 − 2) AB D C
56. 56. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB D C
57. 57. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB = 49+ 4 D C
58. 58. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB = 49+ 4 = 53 D C
59. 59. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB = 49+ 4 = 53 units D C
60. 60. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB = 49+ 4 = 53 units D CD = (−2 −5)2 +(−1−1)2 C
61. 61. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB = 49+ 4 = 53 units D 2 2 2 2 CD = (−2 −5) +(−1−1) = (−7) +(−2) C
62. 62. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB = 49+ 4 = 53 units D 2 2 2 2 CD = (−2 −5) +(−1−1) = (−7) +(−2) C = 49+ 4
63. 63. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB = 49+ 4 = 53 units D 2 2 2 2 CD = (−2 −5) +(−1−1) = (−7) +(−2) C = 49+ 4 = 53
64. 64. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 AB = (2 −(−5)) +(4 − 2) = (7) +(2) AB = 49+ 4 = 53 units D 2 2 2 2 CD = (−2 −5) +(−1−1) = (−7) +(−2) C = 49+ 4 = 53 units
65. 65. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. AB D C
66. 66. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 BC = (−5−(−2)) +(2 −(−1)) AB D C
67. 67. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB D C
68. 68. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 D C
69. 69. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 = 18 D C
70. 70. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 = 18 units D C
71. 71. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 = 18 units D 2 2 AD = (2 −5) +(4 −1) C
72. 72. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 = 18 units D 2 2 2 2 AD = (2 −5) +(4 −1) = (−3) +(3) C
73. 73. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 = 18 units D 2 2 2 2 AD = (2 −5) +(4 −1) = (−3) +(3) C = 9+ 9
74. 74. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 = 18 units D 2 2 2 2 AD = (2 −5) +(4 −1) = (−3) +(3) C = 9+ 9 = 18
75. 75. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 = 18 units D 2 2 2 2 AD = (2 −5) +(4 −1) = (−3) +(3) C = 9+ 9 = 18 units
76. 76. Example 3 The vertices of quadrilateral ABCD are A = (2, 4), B = (-5, 2), C = (-2, -1), and D = (5, 1). b. Use distances to justify your guess. 2 2 2 2 BC = (−5−(−2)) +(2 −(−1)) = (−3) +(3) AB = 9+ 9 = 18 units D 2 2 2 2 AD = (2 −5) +(4 −1) = (−3) +(3) C = 9+ 9 = 18 units It is a parallelogram, as opposite sides are equal.
77. 77. Problem Set
78. 78. Problem Set p. 246 #1-33 odd, 18, 34, 36“If I have seen further it is by standing on the shoulders of giants.” - Isaac Newton

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Distance in the Coordinate Plane

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