Midpoint Formula

The midpoint formula averages corresponding coordinates to locate the point exactly halfway between two endpoints.

A segment’s horizontal and vertical changes form a right triangle for distance and midpoint reasoning. Two coordinate points are joined by a segment with dashed rise and run legs. The midpoint is marked halfway along the segment, connecting coordinate averages with the Pythagorean distance calculation.

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

Add the two $x$-coordinates and divide by $2$.
Add the two $y$-coordinates and divide by $2$.
Keep those results in the order $(x,y)$.
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.

Finding a missing endpoint

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

B(2m_x-x_1,\;2m_y-y_1).

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 $1$.

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?