Geometry · Unit 2
From two points, find the slope, the distance, and the midpoint. From one endpoint and the midpoint, find the other endpoint.
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Slope
Call the first point (x₁, y₁) and the second point (x₂, y₂).
Example: through xy(1,2) and xy(4,8).
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 8 − 2
y₂ − y₁ = 6
x₂ − x₁ = 4 − 1
x₂ − x₁ = 3
m = 63
m = 2
Distance
Example: xy(0,0) to xy(6,8).
d =
x₂ − x₁ = 6 − 0
x₂ − x₁ = 6
y₂ − y₁ = 8 − 0
y₂ − y₁ = 8
d =
d =
d =
d = 10
Midpoint
Example: xy(2,3) and xy(8,11).
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(2 + 82,3 + 112)
midpoint = xy(102,142)
midpoint = xy(5,7)
Two points. Find the slope.
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 7 − 1
y₂ − y₁ = 6
x₂ − x₁ = 4 − 1
x₂ − x₁ = 3
m = 63
m = 2
Answer: 2
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 11 − 3
y₂ − y₁ = 8
x₂ − x₁ = 6 − 2
x₂ − x₁ = 4
m = 84
m = 2
Answer: 2
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 1 − 9
y₂ − y₁ = −8
x₂ − x₁ = 5 − 1
x₂ − x₁ = 4
m = −84
m = −2
Answer: −2
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 4 − 4
y₂ − y₁ = 0
x₂ − x₁ = 3 − (−2)
x₂ − x₁ = 5
m = 05
m = 0
Answer: 0
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 6 − (−1)
y₂ − y₁ = 7
x₂ − x₁ = 7 − 7
x₂ − x₁ = 0
m = 70
The slope is undefined.
Answer: undefined
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 10 − (−2)
y₂ − y₁ = 12
x₂ − x₁ = 1 − (−3)
x₂ − x₁ = 4
m = 124
m = 3
Answer: 3
Two points. Find the midpoint.
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(0 + 62,1 + 92)
midpoint = xy(62,102)
midpoint = xy(3,5)
Answer: xy(3,5)
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(1 + 72,2 + 82)
midpoint = xy(82,102)
midpoint = xy(4,5)
Answer: xy(4,5)
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(−4 + 22,3 + 112)
midpoint = xy(−22,142)
midpoint = xy(−1,7)
Answer: xy(−1,7)
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(0 + 82,−6 + 22)
midpoint = xy(82,−42)
midpoint = xy(4,−2)
Answer: xy(4,−2)
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(5 + 52,4 + 122)
midpoint = xy(102,162)
midpoint = xy(5,8)
Answer: xy(5,8)
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(−8 + 42,1 + 12)
midpoint = xy(−42,22)
midpoint = xy(−2,1)
Answer: xy(−2,1)
Two points. Find the distance.
d =
x₂ − x₁ = 4 − 1
x₂ − x₁ = 3
y₂ − y₁ = 6 − 2
y₂ − y₁ = 4
d =
d =
d =
d = 5
Answer: 5
d =
x₂ − x₁ = 5 − 0
x₂ − x₁ = 5
y₂ − y₁ = 12 − 0
y₂ − y₁ = 12
d =
d =
d =
d = 13
Answer: 13
d =
x₂ − x₁ = 7 − 1
x₂ − x₁ = 6
y₂ − y₁ = 9 − 1
y₂ − y₁ = 8
d =
d =
d =
d = 10
Answer: 10
d =
x₂ − x₁ = 1 − (−2)
x₂ − x₁ = 3
y₂ − y₁ = 8 − 4
y₂ − y₁ = 4
d =
d =
d =
d = 5
Answer: 5
d =
x₂ − x₁ = 3 − 3
x₂ − x₁ = 0
y₂ − y₁ = 9 − 0
y₂ − y₁ = 9
d =
d =
d =
d = 9
Answer: 9
d =
x₂ − x₁ = 7 − 2
x₂ − x₁ = 5
y₂ − y₁ = 11 − (−1)
y₂ − y₁ = 12
d =
d =
d =
d = 13
Answer: 13
The coordinates are fractions. Find the slope, the midpoint, and the distance.
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 134 − 14
y₂ − y₁ = 124
y₂ − y₁ = 3
x₂ − x₁ = 92 − 12
x₂ − x₁ = 82
x₂ − x₁ = 4
m = 34
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(12 + 922,14 + 1342)
midpoint = xy(52,722)
midpoint = xy(52,74)
Distance
d =
x₂ − x₁ = 92 − 12
x₂ − x₁ = 82
x₂ − x₁ = 4
y₂ − y₁ = 134 − 14
y₂ − y₁ = 124
y₂ − y₁ = 3
d =
d =
d =
d = 5
Answer: Slope: 34Midpoint: xy(52,74)Distance: 5
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 92 − 12
y₂ − y₁ = 82
y₂ − y₁ = 4
x₂ − x₁ = 103 − 13
x₂ − x₁ = 93
x₂ − x₁ = 3
m = 43
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(13 + 1032,12 + 922)
midpoint = xy(1132,52)
midpoint = xy(116,52)
Distance
d =
x₂ − x₁ = 103 − 13
x₂ − x₁ = 93
x₂ − x₁ = 3
y₂ − y₁ = 92 − 12
y₂ − y₁ = 82
y₂ − y₁ = 4
d =
d =
d =
d = 5
Answer: Slope: 43Midpoint: xy(116,52)Distance: 5
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 12 − 92
y₂ − y₁ = −82
y₂ − y₁ = −4
x₂ − x₁ = 134 − 14
x₂ − x₁ = 124
x₂ − x₁ = 3
m = −43
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(14 + 1342,92 + 122)
midpoint = xy(722,52)
midpoint = xy(74,52)
Distance
d =
x₂ − x₁ = 134 − 14
x₂ − x₁ = 124
x₂ − x₁ = 3
y₂ − y₁ = 12 − 92
y₂ − y₁ = −82
y₂ − y₁ = −4
d =
d =
d =
d = 5
Answer: Slope: −43Midpoint: xy(74,52)Distance: 5
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 113 − (−13)
y₂ − y₁ = 123
y₂ − y₁ = 4
x₂ − x₁ = 52 − 52
x₂ − x₁ = 02
x₂ − x₁ = 0
m = 40
The slope is undefined.
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(52 + 522,−13 + 1132)
midpoint = xy(52,1032)
midpoint = xy(52,53)
Distance
d =
x₂ − x₁ = 52 − 52
x₂ − x₁ = 02
x₂ − x₁ = 0
y₂ − y₁ = 113 − (−13)
y₂ − y₁ = 123
y₂ − y₁ = 4
d =
d =
d =
d = 4
Answer: Slope: undefinedMidpoint: xy(52,53)Distance: 4
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 74 − 74
y₂ − y₁ = 04
y₂ − y₁ = 0
x₂ − x₁ = 52 − (−32)
x₂ − x₁ = 82
x₂ − x₁ = 4
m = 04
m = 0
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(−32 + 522,74 + 742)
midpoint = xy(12,722)
midpoint = xy(12,74)
Distance
d =
x₂ − x₁ = 52 − (−32)
x₂ − x₁ = 82
x₂ − x₁ = 4
y₂ − y₁ = 74 − 74
y₂ − y₁ = 04
y₂ − y₁ = 0
d =
d =
d =
d = 4
Answer: Slope: 0Midpoint: xy(12,74)Distance: 4
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 272 − 32
y₂ − y₁ = 242
y₂ − y₁ = 12
x₂ − x₁ = 265 − 15
x₂ − x₁ = 255
x₂ − x₁ = 5
m = 125
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(15 + 2652,32 + 2722)
midpoint = xy(2752,152)
midpoint = xy(2710,152)
Distance
d =
x₂ − x₁ = 265 − 15
x₂ − x₁ = 255
x₂ − x₁ = 5
y₂ − y₁ = 272 − 32
y₂ − y₁ = 242
y₂ − y₁ = 12
d =
d =
d =
d = 13
Answer: Slope: 125Midpoint: xy(2710,152)Distance: 13
The coordinates are decimals. Find the slope, the midpoint, and the distance.
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 4.5 − 1.5
y₂ − y₁ = 3
x₂ − x₁ = 4.5 − 0.5
x₂ − x₁ = 4
m = 34
m = 0.75
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(0.5 + 4.52,1.5 + 4.52)
midpoint = xy(52,62)
midpoint = xy(2.5,3)
Distance
d =
x₂ − x₁ = 4.5 − 0.5
x₂ − x₁ = 4
y₂ − y₁ = 4.5 − 1.5
y₂ − y₁ = 3
d =
d =
d =
d = 5
Answer: Slope: 0.75Midpoint: xy(2.5,3)Distance: 5
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 3.6 − 0.6
y₂ − y₁ = 3
x₂ − x₁ = 5.2 − 1.2
x₂ − x₁ = 4
m = 34
m = 0.75
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(1.2 + 5.22,0.6 + 3.62)
midpoint = xy(6.42,4.22)
midpoint = xy(3.2,2.1)
Distance
d =
x₂ − x₁ = 5.2 − 1.2
x₂ − x₁ = 4
y₂ − y₁ = 3.6 − 0.6
y₂ − y₁ = 3
d =
d =
d =
d = 5
Answer: Slope: 0.75Midpoint: xy(3.2,2.1)Distance: 5
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 4.2 − 7.2
y₂ − y₁ = −3
x₂ − x₁ = 4.8 − 0.8
x₂ − x₁ = 4
m = −34
m = −0.75
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(0.8 + 4.82,7.2 + 4.22)
midpoint = xy(5.62,11.42)
midpoint = xy(2.8,5.7)
Distance
d =
x₂ − x₁ = 4.8 − 0.8
x₂ − x₁ = 4
y₂ − y₁ = 4.2 − 7.2
y₂ − y₁ = −3
d =
d =
d =
d = 5
Answer: Slope: −0.75Midpoint: xy(2.8,5.7)Distance: 5
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 14 − 2
y₂ − y₁ = 12
x₂ − x₁ = 6.5 − 1.5
x₂ − x₁ = 5
m = 125
m = 2.4
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(1.5 + 6.52,2 + 142)
midpoint = xy(82,162)
midpoint = xy(4,8)
Distance
d =
x₂ − x₁ = 6.5 − 1.5
x₂ − x₁ = 5
y₂ − y₁ = 14 − 2
y₂ − y₁ = 12
d =
d =
d =
d = 13
Answer: Slope: 2.4Midpoint: xy(4,8)Distance: 13
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 3.5 − 3.5
y₂ − y₁ = 0
x₂ − x₁ = 2.6 − (−2.4)
x₂ − x₁ = 5
m = 05
m = 0
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(−2.4 + 2.62,3.5 + 3.52)
midpoint = xy(0.22,72)
midpoint = xy(0.1,3.5)
Distance
d =
x₂ − x₁ = 2.6 − (−2.4)
x₂ − x₁ = 5
y₂ − y₁ = 3.5 − 3.5
y₂ − y₁ = 0
d =
d =
d =
d = 5
Answer: Slope: 0Midpoint: xy(0.1,3.5)Distance: 5
Slope
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 6.5 − (−1.5)
y₂ − y₁ = 8
x₂ − x₁ = 6.3 − 6.3
x₂ − x₁ = 0
m = 80
The slope is undefined.
Midpoint
midpoint = xy(x₁ + x₂2,y₁ + y₂2)
midpoint = xy(6.3 + 6.32,−1.5 + 6.52)
midpoint = xy(12.62,52)
midpoint = xy(6.3,2.5)
Distance
d =
x₂ − x₁ = 6.3 − 6.3
x₂ − x₁ = 0
y₂ − y₁ = 6.5 − (−1.5)
y₂ − y₁ = 8
d =
d =
d =
d = 8
Answer: Slope: undefinedMidpoint: xy(6.3,2.5)Distance: 8
One endpoint and the midpoint are given. Find the other endpoint.
x₁ + x₂2 = the x-coordinate of the midpoint
2 + x2 = 6
2 + x = 12
x = 12 − 2
x = 10
y₁ + y₂2 = the y-coordinate of the midpoint
5 + y2 = 3
5 + y = 6
y = 6 − 5
y = 1
The other endpoint is xy(10,1)
Answer: xy(10,1)
x₁ + x₂2 = the x-coordinate of the midpoint
1 + x2 = 4
1 + x = 8
x = 8 − 1
x = 7
y₁ + y₂2 = the y-coordinate of the midpoint
8 + y2 = 3
8 + y = 6
y = 6 − 8
y = −2
The other endpoint is xy(7,−2)
Answer: xy(7,−2)
x₁ + x₂2 = the x-coordinate of the midpoint
−4 + x2 = 1
−4 + x = 2
x = 2 − (−4)
x = 6
y₁ + y₂2 = the y-coordinate of the midpoint
2 + y2 = 6
2 + y = 12
y = 12 − 2
y = 10
The other endpoint is xy(6,10)
Answer: xy(6,10)
x₁ + x₂2 = the x-coordinate of the midpoint
7 + x2 = 2
7 + x = 4
x = 4 − 7
x = −3
y₁ + y₂2 = the y-coordinate of the midpoint
−3 + y2 = 1
−3 + y = 2
y = 2 − (−3)
y = 5
The other endpoint is xy(−3,5)
Answer: xy(−3,5)
x₁ + x₂2 = the x-coordinate of the midpoint
0 + x2 = −5
0 + x = −10
x = −10 − 0
x = −10
y₁ + y₂2 = the y-coordinate of the midpoint
−6 + y2 = −1
−6 + y = −2
y = −2 − (−6)
y = 4
The other endpoint is xy(−10,4)
Answer: xy(−10,4)
x₁ + x₂2 = the x-coordinate of the midpoint
−8 + x2 = −3
−8 + x = −6
x = −6 − (−8)
x = 2
y₁ + y₂2 = the y-coordinate of the midpoint
9 + y2 = 2
9 + y = 4
y = 4 − 9
y = −5
The other endpoint is xy(2,−5)
Answer: xy(2,−5)
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