Finding Pythagorean triples using the Pythagorean Theorem is really interesting!

If you’re not sure what Pythagorean triples are, don’t worry. They are simply three whole numbers ((a, b, c)) that fit this equation:

[ a^2 + b^2 = c^2. ]

Some common examples are (3, 4, 5) and (5, 12, 13).

Let’s explore some ways to find these triples!

Method 1: Using the Pythagorean Theorem

1. Start with the formula: The Pythagorean theorem tells us that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

   This looks like:

   [ c^2 = a^2 + b^2. ]

2. Choose values for (a) and (b): Pick two positive whole numbers for (a) and (b). Then use the formula to find (c):

   [ c = \sqrt{a^2 + b^2} ]

   If (c) is a whole number, then ((a, b, c)) is a Pythagorean triple!

Method 2: Generating Triples

Another fun way to find whole number triples is by using these formulas:

1. Pick any two positive whole numbers (m) and (n) (make sure (m > n)):

   • (a = m^2 - n^2)
   • (b = 2mn)
   • (c = m^2 + n^2)

For example, let’s take (m = 2) and (n = 1):

• (a = 2^2 - 1^2 = 3)
• (b = 2 \cdot 2 \cdot 1 = 4)
• (c = 2^2 + 1^2 = 5)

So, we get the triple (3, 4, 5), which is pretty neat!

Conclusion

Finding Pythagorean triples can be like a fun puzzle!

Using either method, you'll find many interesting sets of numbers that fit the Pythagorean theorem.

Happy number hunting!