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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:

$\frac{(-1+4)}{2}=\frac{3}{2}=1.5$

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:

$\frac{(-4+{x}_{2})}{2}=0.5$

To solve, start by multiplying both sides by 2:

$(-4+{x}_{2})=1$

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
$({x}_{1},{y}_{1})$
and
$({x}_{2},{y}_{2})$
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
$(-2,5)$
and
$(7,7)$
?

First, let's plug the information we have into the formula and
simplify:

$(\frac{-2+7}{2},\frac{5+7}{2})=(2.5,6)$

## 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:

$Q(2,-2)=(\frac{-6+{x}_{2}}{2},\frac{-6+{y}_{2}}{2})$

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+x

_{2}= 25
${x}_{2}=R=29$

c. What are the coordinates of the midpoint between
$\left(2,3\right)$
and
$\left(10,7\right)$
?

$(2+\frac{10}{2},3+\frac{7}{2})$

$(\frac{12}{2},\frac{10}{2})$

$\left(6,5\right)$

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?

$Q(4,-6)=\frac{-4+{x}_{2}}{2},\frac{-2+{y}_{2}}{2}$

$4=-4+\frac{{x}_{2}}{2}$

$8=-8+{x}_{2}$

$12={x}_{2}$

$-6=-2+\frac{{y}_{2}}{2}$

$-12=-2+{y}_{2}$

$-10={y}_{2}$

$\left(12,-10\right)$

## Topics related to the Midpoint Formula

## Flashcards covering the Midpoint Formula

## Practice tests covering the Midpoint Formula

## Master the midpoint formula with Varsity Tutors

Finding the midpoint can be challenging, especially when students
are encountering it for the first time. Students also need a fair
amount of math skills under their belt already to approach these
problems. If there's a weakness in their foundational knowledge, the
midpoint formula might feel beyond their reach. Private tutoring in
math can help. With 1-on-1 attention and sessions dedicated to the
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