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## Geometry (all content)

# Midpoint formula

CCSS.Math:

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

The start color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd of the points left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis 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 left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis.

The start color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd is the point halfway between each of the points:

An expression for the x-coordinate of the start color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd is start color #1fab54, start fraction, x, start subscript, 1, end subscript, plus, x, start subscript, 2, end subscript, divided by, 2, end fraction, end color #1fab54:

Similarly, an expression for the y-coordinate of the start color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd is start color #e07d10, start color #e07d10, start fraction, y, start subscript, 1, end subscript, plus, y, start subscript, 2, end subscript, divided by, 2, end fraction, end color #e07d10, end color #e07d10:

That's it! We derived the following formula for the start color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd!

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?(34 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.(12 votes)

- How would you solve a problem in which you do not know point B but are given the midpoint and point A?(15 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.(28 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.(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)
- 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.(6 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.(10 votes)

- Hi, i was wondering were i could find "how to find one of the endpoints given one endpoint and the midpoint"(5 votes)
- I do not think that there is an article or lesson about this but this is fairly simple. If given the one endpoint and a midpoint (as long as you now which one is which) all you have to do is find the change in x and the change in y of this and reflect it across the midpoint. Simply, all you have to do is reflect one endpoint across the midpoint and you will find the other endpoint. Hope this helps!(1 vote)

- Hi, if you are given one point and the midpoint, how do you find the other point? Is there a formula? I didn't find a video on it, only on finding the midpoint when given two points.(4 votes)
- We have a point 𝑃(𝑥₀, 𝑦₀) and a point 𝑀(𝑥₁, 𝑦₁), and we want to find the point 𝑄(𝑥₂, 𝑦₂) such that 𝑀 is the midpoint of segment 𝑃𝑄.

If we view 𝑀 as the image of 𝑃 under translation 𝑇, we get

𝑇 = ⟨𝑥₁ − 𝑥₀, 𝑦₁ − 𝑦₀⟩

By applying the same translation to 𝑀, we get

𝑄 = (𝑥₁ + (𝑥₁ − 𝑥₀), 𝑦₁ + (𝑦₁ − 𝑦₀)) = (2𝑥₁ − 𝑥₀, 2𝑦₁ − 𝑦₀)

Example:

𝑃 = (8, −10), 𝑀 = (−7, 3) ⇒

⇒ 𝑄 = (2・(−7) − 8, 2・3 − (−10)) = (−22, 16)(1 vote)

- The circumferences of two concentric circles differ by 10 inches. Compute the width of the ring between the two circles. Round off your answer to nearest hundredths place.

send help thank you i've been burning my brain i just want to know how :(((3 votes)- So lets say we have r1 and r2. We want to know what r2-r1 is. So C1-C2=10, C1=2πr1 and C2=2πr2.

Thus, 2πr2-2πr1=10, factor out to get 2π(r2-r1)=10, divide by 2π to get r2-r1=10/(2π). Then put this in calculator and round.(2 votes)

- Can you find a midpoint of three or even more given points? Or can there only be a midpoint for two given points?(2 votes)
- You can have a "midpoint" of more than two points, but it may be called something else. For example, if you have a triangle, the incenter is equal distant from all three points, and you can find it by bisecting all the angles. This means there is not a nice formula like the midpoint formula of a line segment. I think there are limitations on 4 quadrilaterals, some can have a center point between the four points and some do not.(3 votes)