Midpoint Formula
Reference for midpoint formula M = ((x1+x2)/2, (y1+y2)/2) with 2D and 3D examples.
Covers distance formula and uses in geometry, navigation, and graphics.
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)