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Amy Ferguson Moncure

IB AA · 4B Three-Dimensional Coordinate Geometry

Midpoint

The midpoint is the average of the coordinates.

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Key Ideas Easy Medium Hard

M = (x₁ + x₂2, y₁ + y₂2)

What to Remember

In the plane

M = (x₁ + x₂2, y₁ + y₂2)

In space

Average z the same way: z₁ + z₂2

Odd sums

If a sum is odd, the coordinate is a fraction with denominator 2.

Order

Adding in either order gives the same midpoint.

Easy

Easy

Midpoints in the plane, with whole-number coordinates.

Example
Find the midpoint of (2, 6) and (8, −2).

M = (x₁ + x₂2, y₁ + y₂2)

M = (2 + 82, 6 + (−2)2)

M = (102, 42)

M = (5, 2)

Answer: (5, 2)

1
Find the midpoint of (1, 5) and (7, 11).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (1 + 72, 5 + 112)

M = (82, 162)

M = (4, 8)

Answer: (4, 8)

2
Find the midpoint of (0, 0) and (8, −4).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (0 + 82, 0 + (−4)2)

M = (82, −42)

M = (4, −2)

Answer: (4, −2)

3
Find the midpoint of (−3, 2) and (5, 8).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (−3 + 52, 2 + 82)

M = (22, 102)

M = (1, 5)

Answer: (1, 5)

4
Find the midpoint of (2, 9) and (6, 1).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (2 + 62, 9 + 12)

M = (82, 102)

M = (4, 5)

Answer: (4, 5)

5
Find the midpoint of (−4, −1) and (2, 7).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (−4 + 22, −1 + 72)

M = (−22, 62)

M = (−1, 3)

Answer: (−1, 3)

Medium

Medium

One coordinate is a fraction.

Example
Find the midpoint of (1, 4) and (6, 9).

M = (x₁ + x₂2, y₁ + y₂2)

M = (1 + 62, 4 + 92)

M = (72, 132)

M = (72, 132)

Answer: (72, 132)

1
Find the midpoint of (−2, 3) and (5, 6).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (−2 + 52, 3 + 62)

M = (32, 92)

M = (32, 92)

Answer: (32, 92)

2
Find the midpoint of (0, −5) and (7, 2).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (0 + 72, −5 + 22)

M = (72, −32)

M = (72, −32)

Answer: (72, −32)

3
Find the midpoint of (3, 1) and (3, 11).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (3 + 32, 1 + 112)

M = (62, 122)

M = (3, 6)

Answer: (3, 6)

4
Find the midpoint of (−6, 4) and (2, 4).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2)

M = (−6 + 22, 4 + 42)

M = (−42, 82)

M = (−2, 4)

Answer: (−2, 4)

5
Find the midpoint of (1, 2, 3) and (5, 8, −1).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (1 + 52, 2 + 82, 3 + (−1)2)

M = (62, 102, 22)

M = (3, 5, 1)

Answer: (3, 5, 1)

Hard

Hard

Midpoints in three dimensions.

Example
Find the midpoint of (0, 0, 0) and (6, 4, 10).

M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (0 + 62, 0 + 42, 0 + 102)

M = (62, 42, 102)

M = (3, 2, 5)

Answer: (3, 2, 5)

1
Find the midpoint of (−2, 5, 1) and (4, −1, 7).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (−2 + 42, 5 + (−1)2, 1 + 72)

M = (22, 42, 82)

M = (1, 2, 4)

Answer: (1, 2, 4)

2
Find the midpoint of (1, −3, 4) and (7, 5, 0).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (1 + 72, −3 + 52, 4 + 02)

M = (82, 22, 42)

M = (4, 1, 2)

Answer: (4, 1, 2)

3
Find the midpoint of (−4, 2, −2) and (3, 5, 6).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (−4 + 32, 2 + 52, −2 + 62)

M = (−12, 72, 42)

M = (−12, 72, 2)

Answer: (−12, 72, 2)

4
Find the midpoint of (2, 0, −3) and (8, 5, 3).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (2 + 82, 0 + 52, −3 + 32)

M = (102, 52, 02)

M = (5, 52, 0)

Answer: (5, 52, 0)

5
Find the midpoint of (−1, 4, 2) and (5, 1, 8).
Show Answer

M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (−1 + 52, 4 + 12, 2 + 82)

M = (42, 52, 102)

M = (2, 52, 5)

Answer: (2, 52, 5)

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