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## High school geometry

### Course: High school geometry > Unit 6

Lesson 1: Distance and midpoints# 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?(33 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?(14 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.(27 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.(7 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.(5 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)

- 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.(3 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)(2 votes)

- Is perpendicular line=intersecting lines?(0 votes)
- Yes, perpendicular lines are intersecting, they just form a 90 degree angle.(6 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.(2 votes)

- how do you find the mid point when all the three axes' co ordinates are given i.e X,Y,Z axes?(2 votes)