Midpoint Formula

Meeting in the Middle

The midpoint is the point exactly halfway between two points.

On a number line, halfway between $a$ and $b$ is simply their average, $\frac{a + b}{2}$ . On a plane, we do the very same thing, just once for each coordinate.

The midpoint’s $x$ is the average of the two $x$ values. Its $y$ is the average of the two $y$ values. The two axes are handled completely independently.

The Formula

$M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$

Average the $x$ ’s, average the $y$ ‘s. That is the whole idea.

Distance vs Midpoint

It is easy to mix these two up. They ask different questions and use different operations:

Formula	Question	Operation
Distance	how far apart?	subtract, square, root
Midpoint	what point is in the middle?	add, divide by 2

Distance subtracts. Midpoint averages.

Example

Find the midpoint of $(1, 2)$ and $(5, 6)$ .

\begin{aligned} M &= \left( \frac{1 + 5}{2}, \; \frac{2 + 6}{2} \right) \\[0.5em] &= \left( \frac{6}{2}, \; \frac{8}{2} \right) = (3, 4) \end{aligned}

Example With Negatives and Fractions

Find the midpoint of $(-3, 4)$ and $(2, -1)$ .

\begin{aligned} M &= \left( \frac{-3 + 2}{2}, \; \frac{4 + (-1)}{2} \right) \\[0.5em] &= \left( -\frac{1}{2}, \; \frac{3}{2} \right) \end{aligned}

The midpoint does not have to land on nice whole numbers. Averaging often gives fractions, and that is perfectly fine.