To determine whether the given sides \( a = 6 \) and \( c = 9 \) form a Pythagorean triple in the context of a right-angled triangle, we need to identify a potential third side, \( b \), and check if the Pythagorean theorem holds. The Pythagorean theorem states:
\[ a^2 + b^2 = c^2 \]
In this equation:
- \( a \) is one leg of the triangle.
- \( b \) is the other leg of the triangle.
- \( c \) is the hypotenuse (the longest side).
We can first identify which side is the hypotenuse. Since \( c = 9 \) is greater than \( a = 6 \), we can use \( c \) as the hypotenuse and denote \( a \) and \( b \) as the legs.
Now, we need to find \( b \) and check if the relationship holds. Since \( a \) and \( c \) are given:
- We know \( c = 9 \).
- We also know \( a = 6 \).
Now, we can rearrange the Pythagorean theorem to solve for \( b \):
\[ b^2 = c^2 - a^2 \]
Calculating \( c^2 \) and \( a^2 \):
\[ c^2 = 9^2 = 81 \] \[ a^2 = 6^2 = 36 \]
Now substituting these values into the formula for \( b^2 \):
\[ b^2 = 81 - 36 \] \[ b^2 = 45 \]
To find the value of \( b \):
\[ b = \sqrt{45} \approx 6.71 \]
Now we have:
- \( a = 6 \)
- \( b \approx 6.71 \)
- \( c = 9 \)
Next, we check if \( a, b, c \) form a Pythagorean triple by verifying:
\[ 6^2 + (\sqrt{45})^2 = 9^2 \]
Calculating both sides:
Left-hand side:
\[ 6^2 + 45 = 36 + 45 = 81 \]
Right-hand side:
\[ 9^2 = 81 \]
Both sides are equal, confirming that \( 6^2 + (\sqrt{45})^2 = 9^2 \) holds true.
Thus, \( a = 6 \), \( b \approx 6.71 \), and \( c = 9 \) indeed follow the Pythagorean theorem, and we conclude:
Answer: Yes, these values form a Pythagorean triple with \( a = 6 \), \( b \approx 6.71 \), and \( c = 9 \).