# 11.12: Midpoint Formula

**At Grade**Created by: CK-12

**Practice**Midpoint Formula

### The Midpoint Formula

In the last section, you saw how to find the distance between two points. In this section, you will learn how to find the point exactly half way between two points.

Find the coordinates of the point that is in the middle of the line segment connecting the points \begin{align*}A = (-7, -2)\end{align*}

Let’s start by graphing the two points:

We see that to get from point \begin{align*}A\end{align*}

In order to get to the point that is halfway between the two points, it makes sense that we should move half the vertical distance and half the horizontal distance—that is, 3 units down and 5 units to the right from point \begin{align*}A\end{align*}

The midpoint is \begin{align*}M = (-7 +5, -2 - 3) = (-2, -5)\end{align*}

**The Midpoint Formula**

We now want to generalize this method in order to find a formula for the midpoint of a line segment.

Let’s take two general points \begin{align*}A = (x_1, y_1)\end{align*}

We see that to get from \begin{align*}A\end{align*}

In order to get to the half-way point, we need to move \begin{align*}\frac{x_2-x_1}{2}\end{align*}

This simplifies to \begin{align*}M =\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}**Midpoint Formula:**

The midpoint of the line segment connecting the points \begin{align*}(x_1, y_1)\end{align*}

It should hopefully make sense that the midpoint of a line is found by taking the average values of the \begin{align*}x\end{align*}

#### Finding the Midpoint

Find the midpoint between the following points.

Let’s apply the Midpoint Formula: \begin{align*}\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}

a) (-10,2)

the midpoint of (-10, 2) and (3, 5) is \begin{align*}\left(\frac{-10+3}{2}, \frac{2+5}{2}\right)=\left(\frac{-7}{2}, \frac{7}{2}\right)=\underline{\underline{(-3.5,3.5)}}\end{align*}

b) (3,5)

the midpoint of (3, 6) and (7, 6) is \begin{align*}\left(\frac{3+7}{2}, \frac{6+6}{2}\right)=\left(\frac{10}{2}, \frac{12}{2}\right)=\underline{\underline{(5,6)}}\end{align*}

#### Finding Endpoints

A line segment whose midpoint is (2, -6) has an endpoint of (9, -2). What is the other endpoint?

In this problem we know the midpoint and we are looking for the missing endpoint.

The midpoint is (2, -6).

One endpoint is \begin{align*}(x_1, x_2) = (9, -2)\end{align*}

Let’s call the missing point \begin{align*}(x, y)\end{align*}

We know that the \begin{align*}x-\end{align*}

We know that the \begin{align*}y-\end{align*}

\begin{align*}-6=\frac{-2+y_2}{2} \Rightarrow -12=-2+y_2 \Rightarrow y_2=-10\end{align*}

The missing endpoint is **(-5, -10).**

Here’s another way to look at this problem: To get from the endpoint (9, -2) to the midpoint (2, -6), we had to go 7 units left and 4 units down. To get from the midpoint to the other endpoint, then, we would need to go 7 more units left and 4 more units down, which takes us to (-5, -10).

### Example

#### Example 1

Find the midpoint between the points (4, -5) and (-4, 5).

Let’s apply the Midpoint Formula: \begin{align*}\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}

The midpoint of (4, -5) and (-4, 5) is \begin{align*}\left(\frac{4-4}{2}, \frac{-5+5}{2}\right)=\left(\frac{0}{2}, \frac{0}{2}\right)=\underline{\underline{(0,0)}}\end{align*}

### Review

Find the midpoint of the line segment joining the two points.

- (3, -4) and (6, 1)
- (2, -3) and (2, 4)
- (4, -5) and (8, 2)
- (1.8, -3.4) and (-0.4, 1.4)
- (5, -1) and (-4, 0)
- (10, 2) and (2, -4)
- (3, -3) and (2, 5)
- An endpoint of a line segment is (4, 5) and the midpoint of the line segment is (3, -2). Find the other endpoint.
- An endpoint of a line segment is (-10, -2) and the midpoint of the line segment is (0, 4). Find the other endpoint.
- Find a point that is the same distance from (4, 5) as it is from (-2, -1), but is
*not*the midpoint of the line segment connecting them.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 11.12.

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### Image Attributions

Here you'll learn how to use the midpoint formula to find the coordinates of the point that is in the middle of the line segment connecting two given points. You'll also use that formula to find one endpoint of a line segment given its other endpoint and its midpoint.

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