Picture a zip line stretched from one cliff to another over a river. On tour day the river is full of crocodiles, so the guide needs the line’s closest height above the water; if it dips below 20 m anywhere over the river, the trip is off. She can’t measure mid-air, but she knows this: the left cliff is 28 m above the water, the right cliff is 20 m above, the landing is 50 m horizontally from the start, and the water extends only the first 30 m. How close will the line get over the water?

This question launches our chapter. We’ll place the scene on the Cartesian plane, model the cable as a straight line, and use three tools—distance, midpoint, and gradient (slope)—to analyze it. These basics turn pictures into numbers and numbers into decisions. Let’s set up the coordinates and begin.ical tools we have.