Formulas for Computing Point Coordinates
Big Idea: In coordinate geometry, we can use powerful formulas to find exact locations of points — whether it's the midpoint between two points, a point that divides a segment in a given ratio, or the distance between points. These formulas turn geometric problems into algebraic calculations!
1. The Distance Formula
The distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is derived from the Pythagorean Theorem:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
This formula measures the straight-line (Euclidean) distance between any two points in the coordinate plane. The horizontal difference $(x_2 - x_1)$ and vertical difference $(y_2 - y_1)$ form the legs of a right triangle, and the distance is the hypotenuse.