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## Geometry (all content)

### Course: Geometry (all content)>Unit 15

Lesson 1: Distance and midpoints

# Distance formula review

Review the distance formula and how to apply it to solve problems.

## What is the distance formula?

The formula gives the distance between two 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 on the coordinate plane:
square root of, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, left parenthesis, start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared, end square root
It is derived from the Pythagorean theorem.
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 line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle. The hypotenuse of the right triangle is unknown and the side made from the point at x one, y one and x two, y one is labeled x two minus x one. The third side is labeled y two minus y one.

## What problems can I solve with the distance formula?

Given two points on the plane, you can find their distance. For example, let's find the distance between left parenthesis, start color #1fab54, 1, end color #1fab54, comma, start color #e07d10, 2, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, 9, end color #1fab54, comma, start color #e07d10, 8, end color #e07d10, right parenthesis:
\begin{aligned} &\phantom{=}\sqrt{(\greenD{x_2 - x_1})^2 + (\goldD{y_2 - y_1})^2} \\\\ &=\sqrt{(\greenD{9 -1})^2 + (\goldD{8 - 2})^2}\quad\small\gray{\text{Plug in coordinates}} \\\\ &=\sqrt{8^2+6^2} \\\\ &=\sqrt{100} \\\\ &=10 \end{aligned}
Notice: we were careful to put the x-coordinates together and the y-coordinates together and not mix them up.