Comprehensive Reference: Analytic Coordinate Geometry and Linear Formulations
Analytic geometry fuses algebra with Euclidean geometry, representing points as coordinate ordered pairs $(x, y)$ and equations as curves on a Cartesian plane.
1. Derivation of the Distance Formula from the Pythagorean Theorem
Connecting any two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ forms a right-angled triangle with legs of length $|x_2 - x_1|$ and $|y_2 - y_1|$. By the Pythagorean Theorem:
d² = (Δx)² + (Δy)² = (x₂ - x₁)² + (y₂ - y₁)²
Taking the principal non-negative square root yields the celebrated Euclidean distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²).