Upcoming SlideShare
×

# Integrated Math 2 Section 6-1

920 views

Published on

Distance in the Coordinate Plane

Published in: Education, Technology, Business
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Be the first to comment

• Be the first to like this

Views
Total views
920
On SlideShare
0
From Embeds
0
Number of Embeds
29
Actions
Shares
0
8
0
Likes
0
Embeds 0
No embeds

No notes for slide
• ### Integrated Math 2 Section 6-1

1. 1. Chapter 6 Graphing Functions
2. 2. Section 6-1 Distance in the Coordinate Plane
3. 3. Essential Questions How do you use the distance formula to find the distance between two points? How do you use the midpoint formula? Where you’ll see this: Geography, market research, community service, architecture
4. 4. Vocabulary 1. Coordinate Plane: 2. Quadrants: 3. x-axis: 4. y-axis: 5. Ordered Pairs: 6. Origin:
5. 5. Vocabulary 1. Coordinate Plane: Two number lines drawn perpendicular to each other 2. Quadrants: 3. x-axis: 4. y-axis: 5. Ordered Pairs: 6. Origin:
6. 6. Vocabulary 1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points 2. Quadrants: 3. x-axis: 4. y-axis: 5. Ordered Pairs: 6. Origin:
7. 7. Vocabulary 1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points 2. Quadrants: Four areas created by the coordinate plane 3. x-axis: 4. y-axis: 5. Ordered Pairs: 6. Origin:
8. 8. Vocabulary 1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points 2. Quadrants: Four areas created by the coordinate plane 3. x-axis: The horizontal axis on the coordinate plane 4. y-axis: 5. Ordered Pairs: 6. Origin:
9. 9. Vocabulary 1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points 2. Quadrants: Four areas created by the coordinate plane 3. x-axis: The horizontal axis on the coordinate plane 4. y-axis: The vertical axis on the coordinate plane 5. Ordered Pairs: 6. Origin:
10. 10. Vocabulary 1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points 2. Quadrants: Four areas created by the coordinate plane 3. x-axis: The horizontal axis on the coordinate plane 4. y-axis: The vertical axis on the coordinate plane 5. Ordered Pairs: Give us points in the form (x, y) 6. Origin:
11. 11. Vocabulary 1. Coordinate Plane: Two number lines drawn perpendicular to each other; used for graphing points 2. Quadrants: Four areas created by the coordinate plane 3. x-axis: The horizontal axis on the coordinate plane 4. y-axis: The vertical axis on the coordinate plane 5. 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 2 Find the distance between the points (0, 4) and (4, 0).
30. 30. Example 2 Find the distance between the points (0, 4) and (4, 0).
31. 31. Example 2 Find the distance between the points (0, 4) and (4, 0).
32. 32. Example 2 Find the distance between the points (0, 4) and (4, 0).
33. 33. Example 2 Find the distance between the points (0, 4) and (4, 0).
34. 34. Example 2 Find the distance between the points (0, 4) and (4, 0). 2 2 2 a +b =c
35. 35. Example 2 Find 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 2 Find 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 2 Find 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 2 Find 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 2 Find 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 2 Find 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 2 Find 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 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?
48. 48. 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
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? A B
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? A B 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? A B 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? A B 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 appears B 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. A B 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) A B 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) A B 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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. A B 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)) A B 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) A B 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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) A B = 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. Homework
78. 78. Homework p. 246 #1-33 odd, 18, 34, 36 “If I have seen further it is by standing on the shoulders of giants.” - Isaac Newton