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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