Midpoint Formula
Reference for midpoint formula M = ((x₁+x₂)/2, (y₁+y₂)/2).
Covers 2D and 3D midpoints, bisecting line segments, and coordinate geometry proofs.
The Formula
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
The midpoint formula finds the point exactly halfway between two given points.
It simply averages the x-coordinates and the y-coordinates separately.
Variables
| Symbol | Meaning |
|---|---|
| M | The midpoint (the result) |
| (x₁, y₁) | Coordinates of the first point |
| (x₂, y₂) | Coordinates of the second point |
Example 1
Find the midpoint between (2, 4) and (8, 10)
x₁ = 2, y₁ = 4, x₂ = 8, y₂ = 10
M = ((2 + 8) / 2, (4 + 10) / 2)
M = (10 / 2, 14 / 2)
M = (5, 7)
Example 2
Find the midpoint between (-6, 3) and (4, -9)
x₁ = -6, y₁ = 3, x₂ = 4, y₂ = -9
M = ((-6 + 4) / 2, (3 + (-9)) / 2)
M = (-2 / 2, -6 / 2)
M = (-1, -3)
When to Use It
Use the midpoint formula when:
- Finding the center of a line segment
- Locating the center of a circle given two endpoints of a diameter
- Dividing a segment into equal halves for construction problems
- Finding the average position between two data points
Key Notes
- The midpoint is simply the arithmetic mean of each coordinate pair — it generalizes naturally: for n points in any dimension, the centroid (center of mass) is the average of all coordinates, making midpoint a special case with n = 2
- Perpendicular bisector property: the perpendicular bisector of a line segment passes through the midpoint at 90° — intersecting two perpendicular bisectors of a triangle's sides gives the circumcenter (center of the circumscribed circle)
- Extends to 3D without modification: M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) — used in 3D graphics, robotics, and CNC machining to find the midpoint of a tool path or interpolate between keyframes
- Section formula generalizes the midpoint: a point that divides segment AB in ratio m:n has coordinates ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)) — the midpoint is the special case where m = n = 1