Midpoint Formula

If $P_1$$\left( {{x_1},{y_1}} \right)$ and $P_2$$\left( {{x_2},{y_2}} \right)$ are the endpoints of a line segment, it’s midpoint can be found by

$$M = \left( {{x_M},{y_M}} \right) = \left( {{{{x_1} + {x_2}} \over 2},{{{y_1} + {y_2}} \over 2}} \right)$$

where the x-coordinate of the midpoint ${x_M}$ is calculated by taking the average of the x-coordinates of the endpoints $P_1$ and $P_2$

$${x_M} = {{{x_1} + {x_2}} \over 2}$$

while the y-coordinate of the midpoint ${y_M}$ is computed by by taking the average of the y-coordinates of the endpoints $P_1$ and $P_2$

$${y_M} = {{{y_1} + {y_2}} \over 2}$$

Example 1: Find the midpoint $M$ of the line segment whose endpoints are $\left( { – 5,1} \right)$ and $\left( {3,7} \right)$.

STEP 1: Calculate the x-coordinate ${x_M}$ of the midpoint $M$.

$$\eqalign{
{x_M} &= {{{x_1} + {x_2}} \over 2} \cr
&= {{ – 5 + 3} \over 2} \cr
&= {{ – 2} \over 2} \cr
{x_M} &= – 1 \cr} $$

STEP 2: Calculate the y-coordinate ${y_M}$ of the midpoint $M$.

$$\eqalign{
{y_M} &= {{{y_1} + {y_2}} \over 2} \cr
&= {{1 + 7} \over 2} \cr
&= {8 \over 2} \cr
{y_M} &= 4 \cr} $$

STEP 3: Putting them together. The coordinates of the midpoint $M$ is

$$M = \left( {{x_M},{y_M}} \right) = \left( { – 1,4} \right)$$

Example 2: The midpoint of the line segment is $\left( { 2,5} \right)$. Find the coordinates of the other endpoint if one endpoint is $\left( { – 3, 9} \right)$.

We let $\left( {x,y} \right)$ the coordinates of the unknown endpoint. Since we know the coordinates of the midpoint and one of the endpoints, we should be able to set up the equations below.

Here’s the equation to find the x-coordinate of the unknown endpoint:

$$\eqalign{
{{x + \left( { – 3} \right)} \over 2} &= 2 \cr
{{x – 3} \over 2} &= 2 \cr
x – 3 &= 4 \cr} $$

$$\boxed{x = 7}$$

And here’s the setup to find the y-coordinate:

$$\eqalign{
{{y + 9} \over 2} &= 5 \cr
y + 9 &= 10 \cr} $$

$$\boxed{y = 1}$$

Therefore, the other endpoint must have the coordinates $\left( {7,1} \right)$.

To be sure that we got the correct answer, we can perform a quick check. Is $\left( {2,5} \right)$ the midpoint of the the points $\left( { – 3,9} \right)$ and $\left( {7,1} \right)$?

Let’s plug in the coordinates of the endpoints $\left( { – 3,9} \right)$ and $\left( {7,1} \right)$ into the midpoint formula.

$\blacksquare$ Finding the x-coordinate of the midpoint:

$$\eqalign{
{x_M} &= {{{x_1} + {x_2}} \over 2} \cr
&= {{ – 3 + 7} \over 2} \cr
&= {4 \over 2} \cr
{x_M} &= 2 \cr} $$

$\blacksquare$ Finding the y-coordinate of the midpoint:

$$\eqalign{
{y_M} &= {{{y_1} + {y_2}} \over 2} \cr
&= {{9 + 1} \over 2} \cr
&= {{10} \over 2} \cr
{y_M} &= 5 \cr} $$

That’s right! It checks that the midpoint is at $\left( {2,5} \right)$.