The Distance Formula

Plot two points anywhere on a coordinate plane. Draw a horizontal line from one and a vertical line from the other, and they meet at a right angle, forming a right triangle. The straight-line distance you want is the hypotenuse of that triangle. The legs are just the horizontal gap between the x-coordinates and the vertical gap between the y-coordinates. Once you see those two numbers as legs, the Pythagorean theorem a² + b² = c² hands you the answer directly: distance = √((x2-x1)² + (y2-y1)²). There's no new idea here, only an old one applied sideways.

The order you subtract in doesn't matter. If you compute x1-x2 instead of x2-x1, you get the negative of the horizontal gap, but squaring it wipes out the negative sign. Same for the vertical gap. So the formula is forgiving about order as long as you're consistent within each difference, and the final distance always comes out positive, as a length should.

This formula is the first tool in coordinate geometry, and everything after it leans on the same right-triangle picture. The midpoint formula averages the same coordinates instead of subtracting them, slope uses the same rise and run but divides instead of applying Pythagoras, and later the equation of a circle is just the distance formula set equal to a fixed radius. Get comfortable seeing the legs of a hidden right triangle in any pair of points, and the rest of coordinate geometry reads as variations on that one move.