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### Course: Geometry (all content)>Unit 15

Lesson 1: Distance and midpoints

# Midpoint formula review

Review the midpoint formula and how to apply it to solve problems.

## What is the midpoint formula?

The formula gives the midpoint of the points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ in the coordinate plane:
$\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)$

## What problems can I solve with the midpoint formula?

Given two points on the plane, you can find their midpoint. For example, let's find the midpoint of $\left(5,3\right)$ and $\left(1,7\right)$:
$\begin{array}{rl}& \phantom{=}\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)\\ \\ & =\left(\frac{5+1}{2},\frac{3+7}{2}\right)\phantom{\rule{1em}{0ex}}\text{Plug in coordinates}\\ \\ & =\left(3,5\right)\end{array}$
Notice: we were careful to put the $x$-coordinates together and the $y$-coordinates together and not mix them up.

Problem 1
What is the midpoint of $\left(6,2\right)$ and $\left(10,0\right)$?

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• How do you find the endpoint of a line when you are given the other endpoint and the midpoint?
• Great question, and it's definitely a situation we need to be able to solve.

When we are given the Midpoint and an Endpoint, and asked to find the Other Endpoint.

We can still use the concepts from the Midpoint Formula by plugging in the information we do have, because…

we normally find the Midpoint by adding the corresponding coordinates, then dividing by two…

(first endpoint + second endpoint) divided by two = Midpoint

So, we know that…
(x1 + x2)/2 = Midpoint x
and that…
(y1 + y2)/2 = Midpoint y

the given Endpoint (x, y) can be plugged into either the first or second x and y locations because addition is Commutative, (it's rearrangeable), so it doesn't matter which end values are first or second,

and the given Midpoint, (x or y), is placed as equal to the Endpoint addition then division.

Given:
Endpoint (3, 5)
Midpoint (7, 10
)

•Find Other Endpoint
=
(x1 + x2)/2 = Midpoint x
and
(y1 + y2)/2 = Midpoint y
=
(x + 3)/2 = 7
and
(y + 5)/2 = 10

•Let's solve for x first:
(x + 3)/2 = 7
Multiply both sides by 2,
to cancel the division on the left.
=
2((x + 3)/2) = 7 • 2
=
x + 3 = 14
Subtract 3 from both sides,
to isolate x on the left.
=
x + 3 -3 = 14 -3
=
x = 11 ←🥳

•Solve for y the same way…
(y + 5)/2 = 10
=
2((y + 5)/2) = 10(2)
=
y + 5 = 20
=
y = 20 -5
=
y = 15 ←🥳

• Other Endpoint is at:
(11, 15) ←🥳🥳

•Each time the pattern of operations will be the same:
Multiply the Midpoint coordinate by 2, then add or subtract the given Endpoint to isolate variable.

2 • (Midpoint x or y) +/- (endpoint)
=
other endpoint

•Be aware if a given Endpoint coordinate is negative, it will be added rather than subtracted to isolate the variable.

(ㆁωㆁ)I hope this helps someone!
• don't u divied when u do the midpoint
• You are finding the average, so one way is to add the two x coordinates and divide by 2, then do the same for the y coordinates.
• I think I might have lost a few braincells in the process
... :[
• Don't worry- you are not the only one X) This was supposed to make me smarter T_T
• how do i find coordinators if im looking for the endpoint
• Are you saying you know one endpoint and the midpoint? If this is what you are asking, find the vector from the endpoint to the midpoint, the starting at midpoint, go along an equivalent vector. For example, endpoint is at (5,6) and midpoint is at (3,8) so the vector is <3-5, 8-6> or <-2,2>. SO starting at (3,8) along vector <-2,2), you end at (1,10).
• formula for the length of scalene triangle
(1 vote)
• what is length triangle mean?
• it means what is the length of the triangle