Find a Missing Point | Math Coach Amy

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Key Ideas Easy Medium Hard

B = 2M − A

What to Remember

Other endpoint

If M is the midpoint of A and B, then each coordinate of B is 2 × (coordinate of M) − (coordinate of A).

Letters

Average the matching coordinates and set them equal to the midpoint. Solve the two equations.

Then the line

Once both points are numbers, the gradient and point-gradient form give the line.

Distance from the origin

d = square root of x² + y² + z²

Easy

Find the other endpoint in the plane.

Example

A is (1, 4). The midpoint of AB is (4, 7). Find B.

The midpoint is the average of the endpoints.

x₁ + x₂2 = 4

x₁ + x₂ = 8

x₂ = 8 − 1

x₂ = 7

y₁ + y₂2 = 7

y₁ + y₂ = 14

y₂ = 14 − 4

y₂ = 10

B = (7, 10)

Answer: (7, 10)

A is (−2, 3). The midpoint of AB is (1, 8). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 1

x₁ + x₂ = 2

x₂ = 2 − (−2)

x₂ = 4

y₁ + y₂2 = 8

y₁ + y₂ = 16

y₂ = 16 − 3

y₂ = 13

B = (4, 13)

Answer: (4, 13)

A is (5, −1). The midpoint of AB is (2, 3). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 2

x₁ + x₂ = 4

x₂ = 4 − 5

x₂ = −1

y₁ + y₂2 = 3

y₁ + y₂ = 6

y₂ = 6 − (−1)

y₂ = 7

B = (−1, 7)

Answer: (−1, 7)

A is (0, 6). The midpoint of AB is (3, 1). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 3

x₁ + x₂ = 6

x₂ = 6 − 0

x₂ = 6

y₁ + y₂2 = 1

y₁ + y₂ = 2

y₂ = 2 − 6

y₂ = −4

B = (6, −4)

Answer: (6, −4)

A is (4, 4). The midpoint of AB is (1, −2). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 1

x₁ + x₂ = 2

x₂ = 2 − 4

x₂ = −2

y₁ + y₂2 = −2

y₁ + y₂ = −4

y₂ = −4 − 4

y₂ = −8

B = (−2, −8)

Answer: (−2, −8)

A is (2, −3, 1). The midpoint of AB is (0, 2, 5). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 0

x₁ + x₂ = 0

x₂ = 0 − 2

x₂ = −2

y₁ + y₂2 = 2

y₁ + y₂ = 4

y₂ = 4 − (−3)

y₂ = 7

z₁ + z₂2 = 5

z₁ + z₂ = 10

z₂ = 10 − 1

z₂ = 9

B = (−2, 7, 9)

Answer: (−2, 7, 9)

Medium

Find the other endpoint in three dimensions.

Example

A is (−1, 4, 2). The midpoint of AB is (3, 1, −2). Find B.

The midpoint is the average of the endpoints.

x₁ + x₂2 = 3

x₁ + x₂ = 6

x₂ = 6 − (−1)

x₂ = 7

y₁ + y₂2 = 1

y₁ + y₂ = 2

y₂ = 2 − 4

y₂ = −2

z₁ + z₂2 = −2

z₁ + z₂ = −4

z₂ = −4 − 2

z₂ = −6

B = (7, −2, −6)

Answer: (7, −2, −6)

A is (6, 0, −4). The midpoint of AB is (2, 5, 1). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 2

x₁ + x₂ = 4

x₂ = 4 − 6

x₂ = −2

y₁ + y₂2 = 5

y₁ + y₂ = 10

y₂ = 10 − 0

y₂ = 10

z₁ + z₂2 = 1

z₁ + z₂ = 2

z₂ = 2 − (−4)

z₂ = 6

B = (−2, 10, 6)

Answer: (−2, 10, 6)

A is (1, 1, 1). The midpoint of AB is (4, −1, 3). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 4

x₁ + x₂ = 8

x₂ = 8 − 1

x₂ = 7

y₁ + y₂2 = −1

y₁ + y₂ = −2

y₂ = −2 − 1

y₂ = −3

z₁ + z₂2 = 3

z₁ + z₂ = 6

z₂ = 6 − 1

z₂ = 5

B = (7, −3, 5)

Answer: (7, −3, 5)

A is (−3, 2, 5). The midpoint of AB is (1, −2, 1). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 1

x₁ + x₂ = 2

x₂ = 2 − (−3)

x₂ = 5

y₁ + y₂2 = −2

y₁ + y₂ = −4

y₂ = −4 − 2

y₂ = −6

z₁ + z₂2 = 1

z₁ + z₂ = 2

z₂ = 2 − 5

z₂ = −3

B = (5, −6, −3)

Answer: (5, −6, −3)

A is (4, −2, 0). The midpoint of AB is (1, 3, 6). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 1

x₁ + x₂ = 2

x₂ = 2 − 4

x₂ = −2

y₁ + y₂2 = 3

y₁ + y₂ = 6

y₂ = 6 − (−2)

y₂ = 8

z₁ + z₂2 = 6

z₁ + z₂ = 12

z₂ = 12 − 0

z₂ = 12

B = (−2, 8, 12)

Answer: (−2, 8, 12)

A is (−5, 1, 3). The midpoint of AB is (1, −3, −1). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 1

x₁ + x₂ = 2

x₂ = 2 − (−5)

x₂ = 7

y₁ + y₂2 = −3

y₁ + y₂ = −6

y₂ = −6 − 1

y₂ = −7

z₁ + z₂2 = −1

z₁ + z₂ = −2

z₂ = −2 − 3

z₂ = −5

B = (7, −7, −5)

Answer: (7, −7, −5)

Hard

Letters in the coordinates, then a distance.

Example

P is (2a + 1, a) and Q is (5 − b, b + 3). The midpoint of PQ is (72, 4). Find a and b.

(2a + 1) + (5 − b)2 = 72

2a − b + 6 = 7

2a − b = 1

a + (b + 3)2 = 4

a + b + 3 = 8

a + b = 5

b = 5 − a

2a − (5 − a) = 1

3a − 5 = 1

3a = 6

a = 2

b = 3

Answer: a = 2, b = 3

Using a = 2 and b = 3 from the previous result is not assumed here. P(5, 2) and Q(2, 6) are the endpoints. Find the equation of the line PQ in the form ax + by + c = 0.

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m = y₂ − y₁x₂ − x₁

m = 6 − 22 − 5

m = 4−3

y − y₁ = m(x − x₁)

y − 2 = −43(x − 5)

y − 2 = −43x + 203

y = −43x + 263

4x + 3y − 26 = 0

Answer: 4x + 3y − 26 = 0

Find the midpoint of (2, −4, 6) and (4, 2, −2), then find the distance from that midpoint to the origin.

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M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (2 + 42, −4 + 22, 6 + (−2)2)

M = (62, −22, 42)

M = (3, −1, 2)

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (3 − 0)² + (−1 − 0)² + (2 − 0)²

d = square root of 9 + 1 + 4

d = square root of 14

Answer: midpoint (3, −1, 2), distance square root of 14

A is (8, −3). The midpoint of AB is (2, 1). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 2

x₁ + x₂ = 4

x₂ = 4 − 8

x₂ = −4

y₁ + y₂2 = 1

y₁ + y₂ = 2

y₂ = 2 − (−3)

y₂ = 5

B = (−4, 5)

Answer: (−4, 5)

A is (3, −2). The midpoint of AB is (1, 4). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 1

x₁ + x₂ = 2

x₂ = 2 − 3

x₂ = −1

y₁ + y₂2 = 4

y₁ + y₂ = 8

y₂ = 8 − (−2)

y₂ = 10

B = (−1, 10)

Answer: (−1, 10)

A is (0, 5, −1). The midpoint of AB is (2, 1, 3). Find B.

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The midpoint is the average of the endpoints.

x₁ + x₂2 = 2

x₁ + x₂ = 4

x₂ = 4 − 0

x₂ = 4

y₁ + y₂2 = 1

y₁ + y₂ = 2

y₂ = 2 − 5

y₂ = −3

z₁ + z₂2 = 3

z₁ + z₂ = 6

z₂ = 6 − (−1)

z₂ = 7

B = (4, −3, 7)

Answer: (4, −3, 7)

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