Midpoint Formula

Midpoint between Two Points

The midpoint between two points with the same $x$- or $y$-coordinate is the midpoint of a horizontal or vertical line segment connecting those points.
The midpoint of a line segment connecting two points divides the line segment into two equal halves. The midpoint between two points on a number line is the average value of the two points.
On the coordinate plane, the same principle is applied when the $x$-coordinates or the $y$-coordinates of two points are equal.

The Midpoint between Points with an Identical Coordinate

Point $A$ $(5, 11)$ and Point $B$ $(5, -4)$ Point $A$ $(-3, 3)$ and Point $B$ $(5, 3)$
The $y$-coordinate of the midpoint is the average of the $y$-values. The $x$-coordinate of the midpoint is the average of the $x$-values.
\begin{aligned} (x,y)&=\left(5,\frac{11+(-4)}{2}\right)\\ &=\left(5, \frac{11-4}{2}\right)\\&=\left(5, \frac{7}{2}\right) \\ &=(5, 3.5)\end{aligned}
\begin{aligned}(x,y)&=\left(\frac{-3+5}{2},3\right)\\ &=\left(\frac{2}{2}, 3\right)\\&=(1,3)\end{aligned}

Applying the Midpoint Formula

The midpoint formula can be used to find the midpoint between any two points in the coordinate plane.

The midpoint formula is used to calculate the coordinates of the midpoint between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane.

The $x$-coordinate of point $M$ is the average of the $x$-coordinates of the two points.
$x=\frac{x_1+x_2}{2}$
The $y$-coordinate of point $M$ is the average of the $y$-coordinates of the two points.
$y=\frac{y_1+y_2}{2}$
The result is the midpoint formula:
$(x,y)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
Step-By-Step Example
Using the Midpoint Formula
Determine the midpoint between point $A$ $(-3, 3)$ and point $B$ $(9, -2)$.
Step 1
Substitute the coordinates of the points in the midpoint formula.
\begin{aligned}(x,y)&=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ &=\left(\frac{-3+9}{2},\frac{3+(-2)}{2}\right)\end{aligned}
Step 2
Simplify.
\begin{aligned}(x,y)&=\left(\frac{-3+9}{2},\frac{3+(-2)}{2}\right)\\&=\left(\frac{6}{2},\frac{1}{2}\right)\\&=\left(3,\frac{1}{2}\right)\end{aligned}
Solution
The midpoint between point $A$ $(-3,3)$ and point $B$ $(9,-2)$ is $\left(3,\frac{1}{2}\right)$.