To determine if the sides \(a = 6\) and \(c = 9\) of a right-angled triangle form a Pythagorean triple, we first need to identify which sides correspond to the definitions of a Pythagorean triple. In a right-angled triangle, the Pythagorean theorem states:
\[ c^2 = a^2 + b^2 \]
where \(c\) is the length of the hypotenuse (the longest side) and \(a\) and \(b\) are the lengths of the other two sides.
Given:
- \(a = 6\) (one of the legs),
- \(c = 9\) (the hypotenuse since it is the longest side).
Next, we need to determine the value of the other leg \(b\). We can reformulate the Pythagorean theorem to solve for \(b\):
\[ b^2 = c^2 - a^2 \]
Now, let's calculate \(c^2\) and \(a^2\):
\[ c^2 = 9^2 = 81 \] \[ a^2 = 6^2 = 36 \]
Now, substituting these values into the equation for \(b^2\):
\[ b^2 = 81 - 36 = 45 \]
To find \(b\), we take the square root of \(45\):
\[ b = \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \]
Now we have the sides of the triangle as \(a = 6\), \(b = 3\sqrt{5}\), and \(c = 9\).
To check if these values form a Pythagorean triple, we need to verify if:
\[ c^2 = a^2 + b^2 \]
Calculating \(b^2\):
\[ b^2 = (3\sqrt{5})^2 = 9 \times 5 = 45 \]
Now check the Pythagorean theorem:
\[ a^2 + b^2 = 36 + 45 = 81 \] \[ c^2 = 81 \]
Since \(c^2 = a^2 + b^2\) holds true, the values do indeed form a Pythagorean triple.
Conclusion:
The sides \(a = 6\), \(b = 3\sqrt{5}\), and \(c = 9\) satisfy the Pythagorean theorem, confirming that they form a Pythagorean triple. Thus, the answer is yes, these values form a Pythagorean triple.