Endpoint Calculator (find the other endpoint from one endpoint and the midpoint)

Understanding the endpoint calculator — concept and principle

This calculator finds the missing endpoint of a line segment when you know one endpoint and the midpoint. It also gives the length of the segment and half its length. Use it for coordinate geometry homework, mapping and design layouts.

How it is calculated

Midpoint formula: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

Solved for the other endpoint: x₂ = 2xₘ − x₁, y₂ = 2yₘ − y₁

Length AB = √((x₂ − x₁)² + (y₂ − y₁)²); AM = MB = AB ÷ 2

The principle

The midpoint is exactly halfway, so the step from A to M is the same as the step from M to B. Add that step to M once more and you reach B. In coordinates, the step is (xₘ − x₁, yₘ − y₁), and B = M + step = (2xₘ − x₁, 2yₘ − y₁). The length follows from the distance formula, which is the Pythagorean theorem on the horizontal and vertical differences.

Worked example

Default inputs: A = (2, −1), M = (5, 3).

1. x₂ = 2 × 5 − 2 = 8.
2. y₂ = 2 × 3 − (−1) = 7.
3. Other endpoint B = (8, 7).
4. Length AB = √(6² + 8²) = √100 = 10.
5. Length AM = MB = 5.

How to read the result

Check that M is really the midpoint: ((2 + 8) ÷ 2, (−1 + 7) ÷ 2) = (5, 3). The step from A to M here is (3, 4), a 3–4–5 right triangle, which is why the half-length is exactly 5.

Common mistakes

• Averaging A and M instead of doubling M and subtracting A.
• Dropping the sign when the known coordinate is negative.
• Mixing up which point is the midpoint.
• Using the midpoint formula in 3D without adding the z-coordinate.

Frequently asked questions

How do you find an endpoint from a midpoint?

Double the midpoint coordinates and subtract the known endpoint: x₂ = 2xₘ − x₁, y₂ = 2yₘ − y₁.

Does it work in three dimensions?

Yes, apply the same rule to z as well.

How long is the segment?

Use the distance formula on the two endpoints; here it is 10.

Sources: OpenStax, College Algebra 2e (2021), §2.1 The Rectangular Coordinate Systems and Graphs