Midpoint Formula
Find the midpoint between two points using M = ((x₁+x₂)/2, (y₁+y₂)/2).
Includes 2D and 3D worked examples.
The Formula
The midpoint formula finds the exact center point between two given points. It works by averaging the x-coordinates and averaging the y-coordinates separately.
Think of it as finding the balance point of a line segment. If you placed the segment on a pin at the midpoint, it would balance perfectly.
The formula is simple but has many practical applications — from construction (finding center points for drilling) to computer graphics (calculating bisection points for rendering).
3D Midpoint
In three dimensions, simply average the z-coordinates as well. This is used in 3D modeling, physics simulations, and spatial analysis.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| M | Midpoint coordinates | Same as input coordinates |
| (x₁, y₁) | First point | — |
| (x₂, y₂) | Second point | — |
Example 1
Find the midpoint between A(2, 8) and B(10, 4).
Average x-coordinates: (2 + 10) / 2 = 12 / 2 = 6
Average y-coordinates: (8 + 4) / 2 = 12 / 2 = 6
M = (6, 6)
Example 2
A bridge spans from support A at (−50, 0) to support B at (150, 0). Where should the center support be placed?
Average x: (−50 + 150) / 2 = 100 / 2 = 50
Average y: (0 + 0) / 2 = 0
M = (50, 0) — the center support goes at position 50 along the span
Example 3
Find the midpoint between two 3D points: P₁(1, 5, 3) and P₂(7, −1, 9).
Average x: (1 + 7) / 2 = 4
Average y: (5 + (−1)) / 2 = 2
Average z: (3 + 9) / 2 = 6
M = (4, 2, 6)
Finding an Endpoint
If you know one endpoint and the midpoint, you can find the other endpoint:
- x₂ = 2 × M_x − x₁
- y₂ = 2 × M_y − y₁
When to Use It
- Construction — finding center points for beams, supports, or holes
- Navigation — determining the halfway point of a journey
- Computer graphics — bisection algorithms and mesh subdivision
- Geometry proofs involving line segment bisectors
- Finding the center of a diameter (which gives the center of a circle)