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### Course: High school geometry > Unit 6

Lesson 1: Distance and midpoints# Midpoint formula

Walk through writing a general formula for the midpoint between two points.

The ${\text{midpoint}}$ of the points $({{x}_{1}},{{y}_{1}})$ and $({{x}_{2}},{{y}_{2}})$ is given by the following formula:

In this article, we're going to derive this formula!

## Deriving the midpoint formula

Let's start by plotting the points $({{x}_{1}},{{y}_{1}})$ and $({{x}_{2}},{{y}_{2}})$ .

The ${\text{midpoint}}$ is the point halfway between each of the points:

An expression for the $x$ -coordinate of the ${\text{midpoint}}$ is ${{\displaystyle \frac{{x}_{1}+{x}_{2}}{2}}}$ :

Similarly, an expression for the $y$ -coordinate of the ${\text{midpoint}}$ is ${{\displaystyle \frac{{y}_{1}+{y}_{2}}{2}}}$ :

That's it! We derived the following formula for the ${\text{midpoint}}$ !

Interestingly, a lot of people don't memorize this exact formula. Instead, they remember that to find the midpoint, you take the average of the $x$ -coordinates and the average of the $y$ -coordinates.

## Practice problem

## Want to join the conversation?

- Good Day! What if the given are the other endpoint and the midpoint? How do you get the coordinates of the other endpoint?(39 votes)
- I believe you would simply find the differences in x and y from the midpoint to the one endpoint, multiply them by two (giving yourself the two side lengths of a right triangle, if you choose to think about your two points in that way), and add these displacements to your given endpoint.(19 votes)

- How would you solve a problem in which you do not know point B but are given the midpoint and point A?(16 votes)
- Here's what I did when I first learned Geometry 50 years ago.

I ask myself,

* How much did I move in what direction from point A to the MIDPOINT.

* Make the same movement in the same directions to find point B.

For example, if point A is at (3,2) and the midpoint is at (-2,5), i would move 5 left and 3 up from A to M.

Do the same thing again. Move 5 left and 3 up from M and you will find yourself at B.(33 votes)

- the line y=x and the curve y=4x-x^2 intersect at the point p and q. find the co-ordinates of p and q(6 votes)
- Substitute for y to get x = 4x - x^2 then move everything to one side, factor, and solve. Try it and if you still need help, ask again.(10 votes)

- Why do we add instead of subtract the numbers(6 votes)
- You basically are averaging the X and Y values.

Consider if you had a grade of 60 and a grade of 100, how would you find the grade that is halfway between them? You would average them. The Midpoint Formula does the same thing.

If one X-value is at 2 and the other X-value is at 8, to find the X-value halfway between them, you add 2+8 and divide by 2 = 5.

Your would repeat the process for the Y-values to find the Y-coordinate of the midpoint.(8 votes)

- Let's say for example, points A and B are divided into four points. How do you get the coordinate of each point?(5 votes)
- It doesn't tell us how to solve for when you have one point and the midpoint... what do you do for that? Because it only shows solving for the midpoint, but what if you're solving for the other point on the line?(6 votes)
- Copy from Kim Seidel's answer:

Find the change in Y and change in X between that 2 points that you have. Your point b will be on the opposite side of the midpoint from point a. And it will have the same change in Y and change in X. For example:

If a = (2,5) and the midpoint = (-1,3):

Change in Y = 5-3 = 2

Change in X = 2-(-1) = 3

Point a is to the right of the midpoint. So point b must be to the left. So, we move in the opposite direction from the midpoint. To go left, we need to change the values to negatives.

y-value of b = 3-2 = 1

x-value of b = -1-3 = -4

b = (-4,1)(4 votes)

- do u guys think this is easy? is this the easiest thing in geometry, im trying to learn b4 i go to school.(2 votes)
- I think that any math that involves a formula is easy.(9 votes)

- I am so confused but okay(4 votes)
- The midpoint formula is basically an average. You add the two x-values and divide by 2. You add the two y-values and divide by 2. This gives you the coordinates of the midpoint (the point located half-way between the original two points).(4 votes)

- Quite simple to me now.(5 votes)
- Hi, i was wondering were i could find "how to find one of the endpoints given one endpoint and the midpoint"(5 votes)
- Look at each pair of coordinates (x, y) and then the other pair of coordinates, 2nd pair(x,y).

now add both the x's and divide by two. (Example: x1+x2/2 = x-midpoint coordinate)

take your answer for that and write it in the first column (here, )

now add the y's and divide by two. (Example: y1+y2/2 = y-midpoint coordinate)

now take that answer and write it in the second column( ,here)

Finally, put the two together and you should have your MIDPOINT. Woo! (x-midpoint, y-midpoint) = the midpoint between the two coordinates you were given in the problem to solve.

this should help!(1 vote)