Distance and Midpoint Formulas

What You’ll Learn

In this lesson you’ll learn the distance formula and the midpoint formula and how to use them to solve problems on the coordinate plane.

When working with points on a coordinate plane, we often need to find:

Distance Formula (derived from the Pythagorean Theorem):

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Midpoint Formula:

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

These formulas work for any two points (x₁, y₁) and (x₂, y₂).

1. Find the distance between points A(2, 3) and B(5, 7).

d = \sqrt{(5-2)^2 + (7-3)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

2. Find the midpoint of points C(−4, 1) and D(6, 9).

M = \left( \frac{-4 + 6}{2}, \frac{1 + 9}{2} \right) = \left( \frac{2}{2}, \frac{10}{2} \right) = (1, 5)

3. Points E(0, 0) and F(8, 6). What is the distance?

d = \sqrt{(8-0)^2 + (6-0)^2} = \sqrt{64 + 36} = \sqrt{100} = 10

These formulas are very practical:

Distance: How far is it from your house to the store on a map? How long is a straight fence between two posts?
Midpoint: Where should you place a support beam in the middle of a bridge? Where is the center of a rectangular garden?

Example: You and a friend live at coordinates (2, 3) and (8, 11) on a city grid. The midpoint tells you exactly where to meet:

\left( \frac{2+8}{2}, \frac{3+11}{2} \right) = (5, 7)

What is the distance between points $(1, 2)$ and $(4, 6)$?

What is the midpoint of points $(-3, 5)$ and $(5, 9)$?

The distance formula comes from which theorem?

Points $A(0,0)$ and $B(0,10)$. What is the distance between them?

Find the midpoint of $(2, -4)$ and $(10, 8)$.