One triple makes many
Show on a big screenThe sides 9, 40 and 41 make a right triangle. What if every side doubles?
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- (9, 40, 41): 81 + 1,600 = 1,681 = 41 × 41.
- Double every side: (18, 80, 82).
- 324 + 6,400 = 6,724 = 82 × 82.
- Every square is 4 times bigger, so the sum still matches.
The rule
Doubling (9, 40, 41) gives (18, 80, 82). That is still a right triangle.
Three whole numbers with a² + b² = c² make a triple. It is also called a Baudhayana triple. Take any whole number k. Multiply every side by k. Each square becomes k² times bigger. So (ka)² + (kb)² = k²a² + k²b². That equals k²(a² + b²), which is k²c². And k²c² = (kc)². The highest common factor is the biggest number that divides every side. A triple whose highest common factor is above 1 is scaled. One whose highest common factor is 1 is primitive.
Rule: 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.
| Triple | Highest common factor | Kind |
|---|---|---|
| (9, 40, 41) | 1 | primitive |
| (18, 80, 82) | 2 | 2 × (9, 40, 41) |
| (27, 120, 123) | 3 | 3 × (9, 40, 41) |
| (11, 60, 61) | 1 | primitive |
A scaled triple comes from a smaller triple. Divide every side by the highest common factor. Divide (18, 80, 82) by 2 to get (9, 40, 41).
Watch out!
Add 1 to each side of (9, 40, 41). You get (10, 41, 42). That is not a triple. 100 + 1,681 = 1,781, but 42 × 42 = 1,764.
Try the lab to finish this lesson.