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11.12: Midpoint Formula

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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*}A=(7,2) and \begin{align*}B = (3, -8)\end{align*}B=(3,8).

Let’s start by graphing the two points:

We see that to get from point \begin{align*}A\end{align*}A to point \begin{align*}B\end{align*}B we move 6 units down and 10 units to the right.

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*}A.

The midpoint is \begin{align*}M = (-7 +5, -2 - 3) = (-2, -5)\end{align*}M=(7+5,23)=(2,5).

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*}A=(x1,y1) and \begin{align*}B = (x_2, y_2)\end{align*}B=(x2,y2) and mark them on the coordinate plane:

We see that to get from \begin{align*}A\end{align*}A to \begin{align*}B\end{align*}B, we move \begin{align*}x_2 - x_1\end{align*}x2x1 units to the right and \begin{align*}y_2 - y_1\end{align*}y2y1 units up.

In order to get to the half-way point, we need to move \begin{align*}\frac{x_2-x_1}{2}\end{align*}x2x12 units to the right and \begin{align*}\frac{y_2-y_1}{2}\end{align*}y2y12 up from point \begin{align*}A\end{align*}A. Thus the midpoint \begin{align*}M\end{align*}M is at \begin{align*}\left(x_1+\frac{x_2-x_1}{2}, y_1+\frac{y_2-y_1}{2}\right)\end{align*}(x1+x2x12,y1+y2y12).

This simplifies to \begin{align*}M =\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}M=(x1+x22,y1+y22). This is the Midpoint Formula:

The midpoint of the line segment connecting the points \begin{align*}(x_1, y_1)\end{align*}(x1,y1) and \begin{align*}(x_2, y_2)\end{align*}(x2,y2) is \begin{align*}\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}(x1+x22,y1+y22).

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*}x and \begin{align*}y-\end{align*}yvalues of the endpoints.

 

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*}(x1+x22,y1+y22)

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*}(10+32,2+52)=(72,72)=(3.5,3.5)

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*}(3+72,6+62)=(102,122)=(5,6)

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*}(x1,x2)=(9,2).

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

We know that the \begin{align*}x-\end{align*}xcoordinate of the midpoint is 2, so: \begin{align*}2=\frac{9+x_2}{2} \Rightarrow 4=9+x_2 \Rightarrow x_2=-5\end{align*}2=9+x224=9+x2x2=5

We know that the \begin{align*}y-\end{align*}ycoordinate of the midpoint is -6, so:

\begin{align*}-6=\frac{-2+y_2}{2} \Rightarrow -12=-2+y_2 \Rightarrow y_2=-10\end{align*}6=2+y2212=2+y2y2=10

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*}(x1+x22,y1+y22)

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.

  1. (3, -4) and (6, 1)
  2. (2, -3) and (2, 4)
  3. (4, -5) and (8, 2)
  4. (1.8, -3.4) and (-0.4, 1.4)
  5. (5, -1) and (-4, 0)
  6. (10, 2) and (2, -4)
  7. (3, -3) and (2, 5)
  8. An endpoint of a line segment is (4, 5) and the midpoint of the line segment is (3, -2). Find the other endpoint.
  9. An endpoint of a line segment is (-10, -2) and the midpoint of the line segment is (0, 4). Find the other endpoint.
  10. 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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Vocabulary

midpoint

The midpoint of two vectors is the location in the center of their endpoints.

Midpoint Formula

The midpoint formula says that for endpoints (x_1, y_1) and (x_2, y_2), the midpoint is @$\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)@$.

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Oct 01, 2012
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Apr 11, 2016
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