Midpoint Formula
Geometry & MeasurementThe midpoint formula finds the exact center point between two given points on the coordinate plane by averaging their coordinates.
Formula
M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)
Definition
The midpoint is the exact middle point of a line segment. The midpoint formula finds it by averaging the x-coordinates and averaging the y-coordinates of the two endpoints.
Example
Midpoint of $(2, 4)$ and $(8, 10)$: $x = (2+8)/2 = 5$, $y = (4+10)/2 = 7$. The midpoint is $(5, 7)$. You can check: the distance from $(2,4)$ to $(5,7)$ equals the distance from $(5,7)$ to $(8,10)$.
Key Insight
Finding the midpoint is like finding the average of two numbers, but done separately for x and y. The midpoint is always halfway between the two points in both directions at once.
Definition
For two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $M = ((x_1+x_2)/2, (y_1+y_2)/2)$ is the point equidistant from both endpoints. It bisects the segment, dividing it into two equal parts. The formula is equivalent to the arithmetic mean of the position vectors.
Example
If $M = (3, 6)$ is the midpoint and one endpoint is $A = (1, 2)$, find the other endpoint $B$: $x_B = 2(3) - 1 = 5$, $y_B = 2(6) - 2 = 10$. So $B = (5, 10)$. This reverse-midpoint technique is tested frequently.
Key Insight
The midpoint is a special case of a weighted average. If you wanted to find the point $1/3$ of the way from $A$ to $B$, you would compute $A + (1/3)(B - A) = (2A + B)/3$. The midpoint is $A + (1/2)(B - A) = (A + B)/2$.
Definition
The midpoint formula is the affine combination $(1/2)p + (1/2)q$ for points $p, q$ in $\mathbb{R}^n$. More generally, the point dividing segment $PQ$ in ratio $m:n$ is $(nP + mQ)/(m+n)$, a weighted average. Midpoints are preserved by affine transformations, which is why the centroid of a triangle (intersection of medians) is an affine invariant.
Example
The centroid of triangle with vertices $A$, $B$, $C$ is $G = (A + B + C)/3$, the average of the three position vectors. This is the triple-midpoint analog and is invariant under all affine maps (translations, scalings, shears).
Key Insight
Barycentric coordinates express any point $P$ in a triangle as a weighted average $P = \alpha A + \beta B + \gamma C$ where $\alpha + \beta + \gamma = 1$. The midpoint formula is the barycentric coordinate $(1/2, 1/2)$ of a segment, a foundational idea in computational geometry and finite element methods.