Testing a corner without a protractor
A builder asks if a corner is square. There is no protractor, only three numbers. A square corner is a right angle, exactly 90°.
- Three corners: (0, 0), (4, 0) and (4, 3).
- One side: its square is 16.
- Another side: its square is 9.
- The longest side: its square is 25.
- 16 + 9 = 25: a square corner.
Try it first. Look at each triangle. Say if a corner is square.
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The rule
Square the length of each of the three sides. A corner is exactly 90° when two squares add to the third. The square corner is opposite the longest side. A near miss does not count.
Take (0, 0), (4, 0) and (4, 3). The three squares are 16, 9 and 25. 16 + 9 = 25. So the corner at (4, 0) is square.
Take (0, 0), (3, 3) and (6, 0) too. The three squares are 18, 18 and 36. 18 + 18 = 36. The corner at (3, 3) is square. No side at that corner lies along an axis.
| The three corners | The three squares | Verdict |
|---|---|---|
| (0, 0), (4, 0), (4, 3) | 16, 9, 25 | square at (4, 0) |
| (0, 0), (3, 3), (6, 0) | 18, 18, 36 | square at (3, 3) |
| (0, 0), (5, 0), (4, 4) | 25, 17, 32 | no square corner |
| (0, 0), (7, 1), (6, 7) | 50, 37, 85 | no square corner |
The last row looks square when drawn. 50 + 37 = 87, not 85. A difference of 2 is still a miss.