Midpoint Formula - Find the Midpoint of a Line Segment

The midpoint formula finds the point halfway between two points on a coordinate plane. If the endpoints are $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint is

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

Average the two $x$-coordinates, then average the two $y$-coordinates. Use this when a problem asks for the point exactly in the middle of a line segment.

Why The Midpoint Formula Works

On a number line, the number halfway between $2$ and $8$ is $\frac{2 + 8}{2} = 5$. The midpoint formula uses that same idea for each coordinate.

First, it finds the horizontal halfway point by averaging $x_1$ and $x_2$. Then it finds the vertical halfway point by averaging $y_1$ and $y_2$. Put those two halfway values together, and you get the point centered between the endpoints.

This works in the coordinate plane because being halfway has to be true in both directions at the same time.

Midpoint Formula Example

Find the midpoint of the segment with endpoints $(-4, 6)$ and $(10, -2)$.

Start with the midpoint formula:

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

Substitute the coordinates:

$$M = \left(\frac{-4 + 10}{2}, \frac{6 + (-2)}{2}\right)$$

Simplify each coordinate:

$$M = \left(\frac{6}{2}, \frac{4}{2}\right) = (3, 2)$$

So the midpoint is $(3, 2)$. A quick check helps: $3$ is halfway between $-4$ and $10$, and $2$ is halfway between $6$ and $-2$.

Common Midpoint Formula Mistakes

One common mistake is adding the coordinates without dividing by $2$. The midpoint is an average, not a sum.

Another mistake is mixing coordinates across axes. You should average the two $x$-values together and the two $y$-values together. Do not combine an $x$-coordinate with a $y$-coordinate.

Sign errors are also common. If one coordinate is negative, keep the sign when you substitute. For example, $6 + (-2)$ is $4$, not $8$.

When To Use The Midpoint Formula

The midpoint formula is useful whenever a problem asks for the center of a segment in the coordinate plane. You see it in coordinate geometry, proofs about bisectors, problems about diagonals of rectangles or parallelograms, and questions where you need to check whether a point lies exactly halfway between two others.

It also connects naturally to the distance formula. The midpoint tells you where the center is, while the distance formula tells you how long the segment is.

Fractional Midpoints Are Still Correct

The formula works for any two points in the coordinate plane. The midpoint does not need integer coordinates. If the averages produce fractions or decimals, that is still correct.

For example, the midpoint of $(1, 2)$ and $(4, 7)$ is

$$\left(\frac{1 + 4}{2}, \frac{2 + 7}{2}\right) = \left(\frac{5}{2}, \frac{9}{2}\right)$$

That midpoint is valid even though neither coordinate is a whole number.

Try A Similar Midpoint Problem

Try finding the midpoint of $(5, -3)$ and $(-1, 9)$. If you want a useful next step, solve it first with the formula and then check on a graph whether your answer looks centered.