How to Find an Endpoint
If you know the midpoint of a segment and one of its endpoints, you can find the other endpoint using simple algebra.
Endpoint Formula
Given an endpoint $A(x_1, y_1)$ and a midpoint $M(x_m, y_m)$, the other endpoint $B(x_2, y_2)$ is:
y₂ = 2yₘ - y₁
Formula Explanation
We use the midpoint formula in reverse. Since $M = \frac{x_1 + x_2}{2}$, we can solve for $x_2$:
$$ y_2 = 2y_m - y_1 $$
Example Problem
Question: Midpoint M is (4, 3) and one endpoint A is (8, 5). Find endpoint B.
Step 1: Identify values.
Midpoint (xₘ, yₘ) = (4, 3)
Endpoint A (x₁, y₁) = (8, 5)
Step 2: Solve for x₂.
x₂ = 2(4) - 8 = 8 - 8 = 0
Step 3: Solve for y₂.
y₂ = 2(3) - 5 = 6 - 5 = 1
Answer: The missing endpoint B is (0, 1).
Frequently Asked Questions
When do I need this calculator?
This is useful when you know the center of an object (or circle) and one outer point, and need to find the opposite outer point. It's common in geometry problems involving circles and diameters.
Does order matter?
No. You can start with either endpoint. If you have Endpoint B and the Midpoint, the formula finds Endpoint A exactly the same way.