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

In words: average the two $x$-coordinates, then average the two $y$-coordinates. Each symbol is one coordinate of one endpoint, and the two $2$'s are what turn each sum into an average. Use this whenever a problem asks for the point exactly in the middle of a line segment.

Why Averaging the Coordinates Works

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

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 $y_2$. Putting those two halfway values together gives 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, which is precisely why you average $x$ with $x$ and $y$ with $y$, never across axes.

Worked Example

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

Start with the formula:

Substitute the coordinates:

Simplify each coordinate:

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

Fractional Midpoints Are Still Correct

The formula works for any two points; 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

valid even though neither coordinate is a whole number.

Practice and Check on a Graph

Find the midpoint of $(5, -3)$ and $(-1, 9)$. Solve it with the formula first, then plot the two endpoints and your answer to confirm the point looks centered between them. That graph check is the fastest way to catch a slip.

Calculation Traps

• Adding without dividing by $2$. The midpoint is an average, not a sum.
• Mixing coordinates across axes. Average the two $x$-values together and the two $y$-values together; never combine an $x$-coordinate with a $y$-coordinate.
• Sign errors. If a coordinate is negative, keep the sign when you substitute. For example, $6 + (-2)$ is $4$, not $8$.

When to Use the Midpoint Formula

Use it whenever a problem asks for the center of a segment in the coordinate plane: coordinate geometry, proofs about bisectors, problems about diagonals of rectangles or parallelograms, and checks of 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.