Exact distances between coordinates

For points A = (x₁,y₁) and B = (x₂,y₂), the displacement is (x₂ − x₁,y₂ − y₁). Pythagoras gives AB² = (x₂ − x₁)² + (y₂ − y₁)². The order of subtraction does not change the distance because the displacements are squared. Keep negative coordinate values inside brackets when subtracting.

Distance is nonnegative and is zero exactly when the points coincide. Simplify a square root by extracting square factors, and retain the exact answer unless a decimal is requested. To compare two lengths, compare their squares when both are nonnegative. This lets you check for equal sides while keeping the square roots exact.

Worked example

Find the exact distance from A = (−1,2) to B = (5,5).

1. The horizontal displacement is 5 − (−1) = 6; the vertical displacement is 5 − 2 = 3.
2. The distance squared is 6² + 3² = 45.
3. Take the nonnegative root and extract the square factor 9.

Answer: AB = √45 = 3√5.