The mathematical foundations of coordinate geometry rest on basic algebraic and geometric principles, making these formulas both intuitive and powerful.
Distance Formula Derivation: For points (x₁,y₁) and (x₂,y₂), form a right triangle with legs of length |x₂-x₁| and |y₂-y₁|. By the Pythagorean theorem, the hypotenuse (our distance) is √[(x₂-x₁)² + (y₂-y₁)²].
Midpoint Formula: The midpoint is simply the average of corresponding coordinates: x-coordinate = (x₁+x₂)/2, y-coordinate = (y₁+y₂)/2.
Slope Formula: Slope represents rise over run, or change in y divided by change in x: m = (y₂-y₁)/(x₂-x₁).
Example: For points A(1,3) and B(7,11), Distance = √[(7-1)² + (11-3)²] = √[36+64] = 10, Midpoint = ((1+7)/2, (3+11)/2) = (4,7), Slope = (11-3)/(7-1) = 8/6 = 4/3.