### 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 $B$ $(5, 3)$ |
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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.

$x=\frac{x_1+x_2}{2}$

$y=\frac{y_1+y_2}{2}$

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

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 $\left(3,\frac{1}{2}\right)$.