Midpoint Formula

The midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ is $M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Why coordinates are averaged

On a number line, the number halfway between $a$ and $b$ is their average $(a+b)/2$. In the coordinate plane, horizontal and vertical positions are averaged independently.

The midpoint’s $x$-coordinate lies halfway between endpoint $x$-values, and its $y$-coordinate lies halfway between endpoint $y$-values.

A reliable method

1. Add the two $x$-coordinates and divide by $2$.
2. Add the two $y$-coordinates and divide by $2$.
3. Keep those results in the order $(x,y)$.
4. Check that the point lies between the endpoints.

Use brackets when adding negative coordinates.

Worked example

Distance from $(3,3)$ to each endpoint is $\sqrt{52}$, confirming it is equally far from both.

Fractions are valid

The midpoint need not have integer coordinates. Between $(2,5)$ and $(7,8)$,

Do not round a midpoint unless the context requires limited measurement precision.

Finding a missing endpoint

If midpoint $M(m_x,m_y)$ and endpoint $A(x_1,y_1)$ are known, the other endpoint is

For midpoint $(4,1)$ and endpoint $(-2,5)$, the other endpoint is $(10,-3)$. Averaging $-2$ and $10$ gives $4$; averaging $5$ and $-3$ gives $1$.

Segment bisectors

A midpoint divides a segment into two congruent pieces. A line, ray, or segment passing through the midpoint is a segment bisector. It need not be perpendicular; a perpendicular bisector has both properties.

Coordinate midpoints can prove that diagonals bisect one another, a defining property of parallelograms.

Partitioning in another ratio

Midpoint is the special $1:1$ division. To find a point a fraction $t$ of the way from $A$ to $B$, use

With $t=1/2$, this becomes the midpoint formula. Other values support trisection and weighted averages.

Three dimensions

Average all three coordinates:

The same component-by-component reasoning applies.

Common mistakes

Crossing coordinate types. Average $x$ with $x$ and $y$ with $y$.

Forgetting division by $2$. A sum is not an average.

Averaging endpoint distances instead of coordinates. Use positions directly.

Rejecting fractional coordinates. Halfway points often fall between grid intersections.

Quick self-check

• Did I preserve coordinate order?
• Are negative values inside brackets?
• Does the result lie between endpoint values in each direction?
• Is it equally distant from both endpoints?

Related topics

• Coordinate Plane
• Distance Formula
• Parallel Lines
• Quadrilaterals