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### Course: High school geometry > Unit 6

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 $({{x}_{1}},{{y}_{1}})$ and $({{x}_{2}},{{y}_{2}})$ in the coordinate plane:

*Want to learn more about the midpoint formula? Check out this video.*

## 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 $({5},{3})$ and $({1},{7})$ :

Notice: we were careful to put the $x$ -coordinates together and the $y$ -coordinates together and not mix them up.

## Want to join the conversation?

- How do you find the endpoint of a line when you are given the other endpoint and the midpoint?(15 votes)
- 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**locations because addition is Commutative, (it's rearrangeable), so it doesn't matter which end values are first or second,*either the first or second*x and y

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**to isolate variable.*add or subtract*the given Endpoint**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.

(**Example in Comments**)

(ㆁωㆁ)I hope this helps someone!(28 votes)

- don't u divied when u do the midpoint(5 votes)
- 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.(15 votes)

- I think I might have lost a few braincells in the process

... :[(9 votes)- Don't worry- you are not the only one X) This was supposed to make me smarter T_T(5 votes)

- how do i find coordinators if im looking for the endpoint(3 votes)
- 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).(10 votes)

- formula for the length of scalene triangle(1 vote)
- what is length triangle mean?(2 votes)
- it means what is the length of the triangle(6 votes)

- How do you find the values of a and b when given the endpoints (1,a) and (b,-4) and the midpoint (4,-6)?(3 votes)
- lowkey brain screaming - im still somewhat confused(1 vote)
- The midpoint formula is just an average.

Add the 2 X-values, then divide by 2.

Add the 2 Y-values, then divide by 2.

You have then found the average for the X and Y values which gives you the point half way between the original 2 points.(5 votes)

- Plot two points to find me square in the above graph. Label the points R and S. A square is a quadrilateral with all for size of the same length. After planting the points trace to square by connecting the points(2 votes)
- how do you find the midpoint of 3 coordinates(2 votes)