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
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
34. Example 2
Find the distance between the points (0, 4) and (4, 0).
2 2 2
a +b =c
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. 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. 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. 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. 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. 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. 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
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.