Integrated Math 1 / Unit 9: Coordinate Geometry and Constructions
9.1 The Distance Formula
Notes
The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. It comes from the Pythagorean theorem.
- Subtract the coordinates, square each difference, add, then take the square root.
- The order of the points does not matter because the differences are squared.
- Leave answers as simple radicals or round when asked.
- Example: $(1, 2)$ to $(4, 6)$: $\displaystyle \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
Practice
What is the distance between $(1, 2)$ and $(4, 6)$?
$\displaystyle \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = 5$.
What is the distance between $(-2, 3)$ and $(4, -5)$?
The differences are 6 and $-8$. Then $\displaystyle \sqrt{36 + 64} = \sqrt{100} = 10$.
What is the distance between $(1, 1)$ and $(3, 5)$?
$\displaystyle \sqrt{2^2 + 4^2} = \sqrt{20} = 2\sqrt{5}$.
Ready for more? Take the Unit 9 test.