A city engineer plans a footbridge across a lake from point X to point Y. She draws a right triangle XYZ with the right angle at X, measures 800 feet from X to Z, and 1,000 feet from Y to Z. How long will the bridge XY be?

1. The right angle is at X, so XY and XZ are the legs and YZ, at 1,000 feet, is the hypotenuse. 2. The Pythagorean Theorem gives a squared plus b squared equals c squared for the legs and hypotenuse. 3. Substitute: XY squared plus 800 squared equals 1,000 squared. 4. Compute the squares: XY squared plus 640,000 equals 1,000,000. 5. Subtract 640,000 from both sides: XY squared equals 360,000 square feet. 6. Take the square root: XY equals 600, so the bridge will be 600 feet long. 7. 1,280 feet treats both measured distances as legs and adds their squares, but 1,000 feet is the hypotenuse and must be squared on its own side. 8. 200 feet subtracts the raw lengths without squaring, which is not how the theorem works. 9. 1,800 feet simply adds the two measurements, which measures the walk around the lake rather than the bridge. _Source: OpenStax Prealgebra (CC BY 4.0), Ch 9 "Math Models and Geometry", section 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem_