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## High school geometry

### 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 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:
left parenthesis, 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, comma, 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, right parenthesis

## 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 first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two.
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:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A point is directly between both points with unknown coordinates.
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:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A point is directly between both points with an x value of x one plus x two all divided by two. Its y coordinate is unknown.
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:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. A tick mark between them is labeled x one plus x two all divided by two. There are two tick marks on the y axis labeled y one and y two. A tick mark between them is labeled y one plus y two all divided by two. There is a point at x one, y one and another point at x two, y two. A point is directly between both points with an x value of x one plus x two all divided by two. Its y coordinate is y one plus y two all divided by two.
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!
left parenthesis, 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, comma, 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, right parenthesis
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

Point start color #6495ed, A, end color #6495ed is at start color #6495ed, left parenthesis, minus, 6, comma, 8, right parenthesis, end color #6495ed and point start color #28ae7b, B, end color #28ae7b is at start color #28ae7b, left parenthesis, 6, comma, minus, 7, right parenthesis, end color #28ae7b.
What is the midpoint of line segment start overline, A, B, end overline?
An x- y- coordinate plane where the x and y tick marks scale by one. A line segment A B has endpoint A at negative six, eight and endpoint B is at six, negative seven.
left parenthesis
,
right parenthesis

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

* 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.
• 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
• 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.
• Let's say for example, points A and B are divided into four points. How do you get the coordinate of each point?
• 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.
• do u guys think this is easy? is this the easiest thing in geometry, im trying to learn b4 i go to school.
• I think that any math that involves a formula is easy.
• 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.
• 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)