How to Find an Endpoint

Learn how to find an endpoint using the midpoint of a segment with a couple of good examples.

Example #1:

The midpoint of a segment is M(2, 3). One endpoint is A(-2, -1). Find the other coordinates of the other endpoint B.

Use the midpoint formula M[(x₁ + x₂)/2, (y₁ + y₂)/2] and let the coordinates of B be (x₂, y₂)

Notice that (x₁ + x₂)/2 is the x-coordinate of the midpoint. So (x₁ + x₂)/2 = 2

Notice that (y₁ + y₂)/2 is the y-coordinate of the midpoint. So (y₁ + y₂)/2 = 3

Let x₁ = -2 and y₁ = -1 and solve the two equations below:

(-2 + x₂)/2 = 2 and (-1 + y₂)/2 = 3

Solve (-1 + y₂)/2 = 3

Multiply both sides of (-1 + y₂)/2 = 3 by 2

2(-1 + y₂)/2 = 3(2)

Simplify

-1 + y₂ = 6

Add 1 to both sides of the equation

-1 + 1 + y₂ = 6 + 1

Simplify

y₂ = 7

Solve (-2 + x₂)/2 = 2

Multiply both sides of (-2 + x₂)/2 = 2 by 2

2(-2 + x₂)/2 = 2(2)

Simplify

-2 + x₂ = 4

Add 2 to both sides of the equation

-2 + 2 + x₂ = 4 + 2

Simplify

x₂ = 6

The coordinates other endpoint B are (6, 7)

Example #2:

The midpoint of a segment is M(-1, 0). One endpoint is A(5, 6). Find the other coordinates of the other endpoint B.

Use the midpoint formula M[(x₁ + x₂)/2, (y₁ + y₂)/2] and let the coordinates of B be (x₂, y₂)

Let x₁ = 5 and y₁ = 6 and solve the two equations below:

(5 + x₂)/2 = -1 and (6 + y₂)/2 = 0

Solve (6 + y₂)/2 = 0

Multiply both sides of (6 + y₂)/2 = 0 by 2

2(6 + y₂)/2 = 0(2)

Simplify

6 + y₂ = 0

Subtract 6 from both sides of the equation

6 - 6 + y₂ = 0 - 6

Simplify

y₂ = -6

Solve (5 + x₂)/2 = -1

Multiply both sides of (5 + x₂)/2 = -1 by 2

2(5 + x₂)/2 = -1(2)

Simplify

5 + x₂ = -2

Subtract 5 from both sides of the equation

5 - 5 + x₂ = -2 - 5 

Simplify

x₂ = -7

The coordinates other endpoint B are (-7, -6)