GPS Navigation Math

Pre-Algebra · Axiom Academy

How your phone turns two places into two pairs of numbers — then uses the distance formula to find the straight-line distance between them. 1. A Place Is Just Two Numbers GPS can't think in street names. It lays a grid over the map and gives every spot an x-coordinate (how far right) and a y-coordinate (how far up). Your phone's blue dot and the place you searched for each become a single point (x, y) — exactly like plotting on a graph. 2. The Distance Formula Is a Right Triangle To get from one point to the other you travel some amount across and some amount up . Those two trips are the legs of a right triangle, and the straight-line distance is its hypotenuse . The horizontal leg is , the vertical leg is , and Pythagoras finishes the job. — how far right (or left) you move. — how far up (or down) you move. Across then up meet at — that's what lets us use Pythagoras. The straight-line distance d — the "as the crow flies" length. Put in numbers. Say you're at (0, 0) and the store is at (3, 4) . Find each leg, square them, add, and take the square root — the same four moves GPS makes for every distance it reports. Adding the legs gives the distance if you walk along the streets — across, then up. But the straight-line distance cuts the corner, so it's always shorter than the two legs: here 5 beats 7. 4. Why the Number Keeps Changing

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