Understanding Pythagorean Triplets
Pythagorean triplets are sets of three positive integers \(a\), \(b\), and \(c\) that satisfy the equation:
\[ a^2 + b^2 = c^2 \]
Here, \(c\) is the largest number (hypotenuse), and \(a\) and \(b\) are the other two sides of a right triangle.

Finding Triplets with 6
When one member of a Pythagorean triplet is 6, we can find the other members using the formula for generating Pythagorean triplets.

Common Triplet Formulas
- **Using the formula:**
For any integers \(m\) and \(n\) (where \(m > n > 0\)):
\[ a = m^2 - n^2 \]
\[ b = 2mn \]
\[ c = m^2 + n^2 \]
We can assume \(a\) or \(b\) is 6 and solve for \(m\) and \(n\).

Examples of Triplets with 6
- **Example 1:** If \(b = 6\):
- Let \(m = 3\) and \(n = 1\):
- \(a = m^2 - n^2 = 3^2 - 1^2 = 9 - 1 = 8\)
- \(b = 2mn = 2 \times 3 \times 1 = 6\)
- \(c = m^2 + n^2 = 3^2 + 1^2 = 9 + 1 = 10\)
- **Triplet:** (8, 6, 10)
- **Example 2:** If \(a = 6\):
- Let \(m = 2\) and \(n = 1\):
- \(a = m^2 - n^2 = 4 - 1 = 3\)
- \(b = 2mn = 4\)
- \(c = m^2 + n^2 = 5\)
- **Triplet:** (6, 8, 10)

Conclusion
The Pythagorean triplet including 6 can be found using various combinations of \(m\) and \(n\). The most common triplet is (6, 8, 10), showcasing the relationship between these numbers in a right triangle.