Distance Formula Discovery

Pre-Algebra · Axiom Academy

How far apart are two points on a grid? Hidden inside the answer is a right triangle you already know how to measure. The distance between two points, in one move Counting squares only works when two points sit in the same row or column. The moment they are diagonal from each other, you are stuck — until you notice that the horizontal gap and the vertical gap form the legs of a right triangle, and the straight-line distance is its hypotenuse. Watch the two points connect: first the horizontal leg, then the vertical leg, then the slanted hypotenuse that joins them. The legs are just and — how far apart the points are across and up. Pythagoras finishes the job. The dashed slant is the real distance — and it is exactly the hypotenuse of the triangle the two gaps make. Move a point — the triangle follows Drag point B anywhere on the grid. The right triangle reshapes itself, and the readout tracks the horizontal gap, the vertical gap, and the distance . The corner stays square no matter where you drop it — that is what keeps Pythagoras in play. As B moves, the same two-step recipe runs every time: square the gaps, add, take the square root. Pick a pair of points. The figure redraws to scale and the distance formula runs on those exact coordinates. The formula is just the Pythagorean theorem written with x 's and y 's — nothing new, only renamed. Order never matters: (5-2) or (2-5) square to the same thing, so A to B is the same distance as B to A .

This is the written version of the interactive lesson above. See the full Pre-Algebra course.