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# Midpoint Formula

The midpoint of a line segment is the point at the very center of a line segment. The line segment is split into two parts of exactly the same length when divided at the midpoint, meaning that the midpoint is always a point that falls on the line segment itself. It can be found in either one dimension (on a number line) or two (on a graph).
But before we learn how to calculate the midpoint of a line segment, it's important to know what a line segment is. A true line in geometry is infinitely long in either direction. A ray is a type of line that has one endpoint and is infinitely long in the other direction. A line segment has two endpoints, making it possible to find the segment's midpoint.

## How to find the midpoint of a segment on a number line

The midpoint formula can be used to find the midpoint or an endpoint of a line segment. On a number line, the midpoint of ${x}_{1}$ and ${x}_{2}$ ( ${x}_{1}$ and ${x}_{2}$ being endpoints) is calculated using this formula:
$\frac{{x}_{1}+{x}_{2}}{2}$
An observant student might notice that the midpoint formula on a number line is the same as taking the average of the endpoints.
Let's look at an example. On this number line, what is the midpoint of -1 and 4?
Let's plug in the two endpoints to the formula:
$\frac{\left(-1+4\right)}{2}=\frac{3}{2}=1.5$
The midpoint is 1.5.

## How to find the endpoint of a segment on a number line

We can also use this formula to find an endpoint if we know both the midpoint and the other endpoint.
If the midpoint of line segment PR is 0.5 and the coordinate of P is -4, what is the coordinate of R? Let's take a look at how to set this up using the same formula as above:
$\frac{\left(-4+{x}_{2}\right)}{2}=0.5$
To solve, start by multiplying both sides by 2:
$\left(-4+{x}_{2}\right)=1$
Now, add 4 to both sides:
${x}_{2}=5$
The coordinate of R is 5.

## How to find the coordinates of a midpoint in two dimensions

If you are given two points in the plane $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ and need to find the point halfway between them, you will need this formula to find the coordinates of the midpoint:
$\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}$
Another way to think about this is that the x coordinate of the midpoint is the average of the x coordinates. The same is true of the y coordinates. Let's look at an example:
What are the coordinates of the midpoint between $\left(-2,5\right)$ and $\left(7,7\right)$ ?
First, let's plug the information we have into the formula and simplify:
$\left(\frac{-2+7}{2},\frac{5+7}{2}\right)=\left(2.5,6\right)$

## How to find the coordinates of an endpoint in two dimensions

To find the endpoint of a line segment, you need to know the midpoint and the other endpoint. For example, if P of the line segment PR has coordinates $\left(-6,-6\right)$ and the midpoint is Q $\left(2,-2\right)$ , what are the coordinates of R?
You'll use the midpoint formula to solve two equations to find the coordinates of R. Set up two equations using the coordinates of P:
$Q\left(2,-2\right)=\left(\frac{-6+{x}_{2}}{2},\frac{-6+{y}_{2}}{2}\right)$
Solve the first equation for the x-coordinate:
$\frac{-6+{x}_{2}}{2}=2$
$-6+{x}_{2}=4$
${x}_{2}=10$
Solve the next equation to find the y-coordinate:
$\frac{-6+{y}_{2}}{2}=-2$
$-6+{y}_{2}=-4$
${y}_{2}=2$
That makes the coordinates of R $\left(10,2\right)$ .

## Midpoint formula practice questions

a. What is the midpoint of the line from -6 to 21 on the number line?
$\frac{-6+21}{2}=\frac{15}{2}=7.5$
b. If the midpoint of line segment PR is 12.5 and the coordinate of P is -4, what is the coordinate of R? Let's take a look at how to set this up using the same formula as above:
$\frac{-4+{x}_{2}}{2}=12.5$
$-4+{x}_{2}=25$
-4+x2 = 25
${x}_{2}=R=29$
c. What are the coordinates of the midpoint between $\left(2,3\right)$ and $\left(10,7\right)$ ?
$\left(2+\frac{10}{2},3+\frac{7}{2}\right)$
$\left(\frac{12}{2},\frac{10}{2}\right)$
$\left(6,5\right)$
d. If P of the line segment PR has coordinates $\left(-4,-2\right)$ and the midpoint is Q $\left(4,-6\right)$ , what are the coordinates of R?
$Q\left(4,-6\right)=\frac{-4+{x}_{2}}{2},\frac{-2+{y}_{2}}{2}$
$4=-4+\frac{{x}_{2}}{2}$
$8=-8+{x}_{2}$
$12={x}_{2}$
$-6=-2+\frac{{y}_{2}}{2}$
$-12=-2+{y}_{2}$
$-10={y}_{2}$
$\left(12,-10\right)$

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