Scaled triples
Multiply every number by k and the triple still works.
Bigger triples from (3, 4, 5)
We know that (3, 4, 5) is a Baudhāyana triple. What about (30, 40, 50)? Every number is 10 times bigger.
And what about (300, 400, 500)?
Do you think these are Baudhāyana triples too?
What this lesson covers
The idea
If (a, b, c) is a Baudhāyana triple, so is its scaled version (ka, kb, kc) for any positive integer k, since (ka)² + (kb)² = k²(a² + b²) = (kc)²; hence there are infinitely many triples.
Bigger triples from (3, 4, 5)
We know that (3, 4, 5) is a Baudhāyana triple. What about (30, 40, 50)? Every number is 10 times bigger.
And what about (300, 400, 500)?
Do you think these are Baudhāyana triples too?
- Yes, both are
- No, the numbers are too big
- Only (30, 40, 50) is
Multiply by k
Choose a triple and a multiplier k. The toy multiplies every number by k and then checks the squares.
A scaled version is a triple too
If (a, b, c) is a Baudhāyana triple, so is its scaled version (ka, kb, kc) for any positive whole number k.
Why? Every square becomes k × k times bigger: (ka)² + (kb)² = k²a² + k²b² = k²(a² + b²) = k²c² = (kc)².
So (3k, 4k, 5k) works for every k. There are infinitely many Baudhāyana triples.
Notes
If (a, b, c) is a Baudhāyana triple, so is (ka, kb, kc) for every positive whole number k. (ka)² + (kb)² = k²(a² + b²) = k²c² = (kc)². So there are infinitely many triples.
Check yourself
Scale (5, 12, 13) by k = 4. What is the new hypotenuse?
Answer: 52
13 × 4 = 52. The new triple is (20, 48, 52).
Which of these is a scaled version of (8, 15, 17)?
A triple (3k, 4k, 5k) has hypotenuse 65. What is k?
Answer: 13
5 × k = 65, so k = 13. The triple is (39, 52, 65).
How many Baudhāyana triples of the form (3k, 4k, 5k) are there?
- (16, 23, 25). That adds 8 to each number. Scaling multiplies each number by the same k: 8, 15, 17 become 8k, 15k, 17k.
- (16, 30, 34) — correct. Yes! Every number is multiplied by k = 2: 16, 30, 34.
- (24, 45, 52). 24 and 45 come from k = 3, but k = 3 makes the last number 51, not 52. Every number needs the same k.
- Infinitely many: k can be any positive whole number and never runs out. — correct. Yes! k = 1, 2, 3, 4, ... gives a new triple each time.
- Only 10: k stops at 10.. k does not stop. (330, 440, 550) is a triple, and so is (3000, 4000, 5000).
- Just one: (3, 4, 5).. Every scaled version is a triple too: (6, 8, 10), (9, 12, 15), and so on.