The big picture
The gate, the ladder and the store room used one idea. It was squares on the sides.
| Idea | How it goes |
|---|---|
| The long side | Always across from the square corner. The hypotenuse. |
| A square on a side | A side of length s carries a square of s × s tiles. |
| The relationship | On a right-angled triangle, a² + b² = c². 9 + 16 = 25. |
| Only at 90° | Under 90°, the two squares at the corner are bigger. Over 90°, the one across is bigger. |
| Long side wanted | Square both short sides, add, then hunt for a match. |
| Short side wanted | Subtract the known short square from the long square, then hunt. |
| Testing three lengths | Do they close? Square the two smaller and add. One apart is still no. |
| Equal sides | Take a right triangle with two equal sides a. Its long side has c² = 2a². |
| A root as a decimal | Square your guess. Too low, guess higher. Too high, guess lower. |
| Scaled triples | Multiply every side of a triple by the same whole number. It is still a triple. A triple with no common factor above 1 is primitive. |
| A hidden depth | The bent stem keeps its length. Write the sides of the right triangle with d, then solve. |
| Using it | A gate brace adds two squares. A ladder takes one square away. |
Squares show whether a corner is a right angle. Add the two small squares and compare with the big one.
Your textbook has this as Chapter 2 of Part II. Read it next.