Distance Formula Application

Algebra 1 · Axiom Academy

From city blocks to coordinate planes — the straight-line distance between two points is just the Pythagorean theorem, put in your hands. You're at A (1, 1) and need to reach B (4, 4) across the city — you can't cut through buildings, so how far is the straight-line flight if you could go directly? Walk 3 blocks east and 3 blocks north to get from A to B. Watch those two legs turn into a right triangle — the straight-line flight is its hypotenuse. Move B — read the distance straight off the legs Drag point B anywhere. The horizontal gap is Δx, the vertical gap is Δy, and the formula just squares them, adds, and square-roots: d = √(Δx² + Δy²). Run it backward — find the missing coordinate C is at (2, 5) and D sits somewhere on the line y = 1. Slide D's x until it's exactly 5 units from C. There's more than one answer — hunt for both. One idea, three moves: feel it (the diagonal is a hypotenuse), solve it (d = √(Δx² + Δy²) for any two points), run it backward (a known distance can pin down a missing coordinate). The same square-and-root distance powers GPS navigation , video-game collision , and robot path-planning — anywhere "how far apart?" needs an answer.

This is the written version of the interactive lesson above. See the full Algebra 1 course.