# Midpoint Formula

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In this video, you will learn the midpoint formula and how to use the midpoint formula to calculate the midpoint of a line segment if the two endpoints are known. Also, you will learn how to calculate the coordinates of an endpoint if the midpoint and the other endpoint is given.

## What is the midpoint formula?

The midpoint formula is based on the average of the x-coordinates and the average of the y-coordinates. The formula is used to find the coordinates of the midpoint of a line segment in the x-y plane.

Midpoint Formula: $(X,Y)=(\frac{X_1 + X_2}{2}, \frac{Y_1 + Y_2}{2})$
In other words, we are simply finding the average of the two x-values and the two y-values.

To better understand how to apply the formula, let’s take a look at a couple of midpoint formula problems.

For example:
Given points $A(1,5)$ and $B(7,3)$, find the midpoint.

By using the midpoint formula, substitute each value into the formulahttps://caddellprep.com/subjects/common-core-geometry/midpoint-formula/?preview=true
$(X,Y)=(\frac{X_1 + X_2}{2}, \frac{Y_1 + Y_2}{2})$
$(X,Y)=(\frac{1+7}{2}, \frac{5+3}{2})$

Combine like terms
$(X,Y)=(\frac{8}{2}, \frac{8}{2})$

Divide each expression
$(X,Y)=(4, 4)$

Let’s try an example where only one endpoint and the midpoint is given. In this problem we will find the other endpoint.

For example:
$M$ is the midpoint of $\overline{AB}$. The coordinates of $B$ are $(4,-3)$ and the coordinates of $M$ are $(-3,-1)$. Find the coordinates of $A$.

Like the previous example, substitute the x and the y-values.
$(X,Y)=(\frac{X_1 + X_2}{2}, \frac{Y_1 + Y_2}{2})$
$(-3,-1)=(\frac{4 + X}{2}, \frac{-3 + Y}{2})$

Separate the equation to solve for x and y individually
x
$-3 =\frac{4 + X}{2}$
$-6 = 4 + x$
$-10 = x$

y
$-1=\frac{-3 + Y}{2}$
$-2 = -3 + y$
$1 = y$

## Video-Lesson Transcript

Let’s go over the midpoint formula.

We have two points – $A$ and $B$ in the $XY$-system.

The midpoint is the point in the middle of these two points.

Remember, $A$ and $B$ is a line segment each with corresponding $x$ and $y$ coordinates. So the midpoint also has $x$ and $y$ coordinates.

Let’s call the coordinates of $A$ as $x_1$ and $y_1$. Likewise, $B$ coordinates are $x_2$ and $y_2$.

The midpoint has the same distance not only between $A$ and $B$ but also between the two $x$-coordinates and $y$-coordinates.

Let’s call the midpoint coordinates as $x_m$ and $y_m$. Midpoint coordinates are the average of the two coordinates.

So our formula is:

$\big( \dfrac{(x_1 + x_2)}{2} , \dfrac{(y_1 + y_2)}{2} \big)$

For example:

Let’s find the midpoint of a line segment which has point $A$ with coordinates $(1, 5)$ and point $B$ with $(7, 3)$ coordinates.

Let’s use our formula – average of the two $X$ coordinates and the two $Y$ coordinates.

$\big( \dfrac{(x_1 + x_2)}{2} , \dfrac{(y_1 + y_2)}{2} \big)$

Let’s label the coordinates first so we won’t be confused.

Then substitute the given

$\big( \dfrac{(1 + 7)}{2} , \dfrac{(5 + 3)}{2} \big)$
$1 + 7 = 8 \div 2 = 4$
$5 + 3 = 8 \div 2 = 4$

So our midpoint is $(4, 4)$

The midpoint of the line segment from point A to point B is found by taking the average of the x-coordinates and the average of the y-coordinates

This is just a coincidence that we came up with an identical value. But we can come up with any value for the $x$– and $y$-coordinates.