To determine whether the given values \( a = 6 \) (one of the legs) and \( c = 9 \) (the hypotenuse) form a Pythagorean triple, we need to find the length of the other leg (let's call it \( b \)) and check if the relationship defined by the Pythagorean theorem holds. The Pythagorean theorem states that for a right-angled triangle:
\[ a^2 + b^2 = c^2 \]
Step 1: Plug in the known values
We know:
- \( a = 6 \)
- \( c = 9 \)
We will solve for \( b \):
\[ 6^2 + b^2 = 9^2 \]
Calculating the squares:
\[ 36 + b^2 = 81 \]
Step 2: Isolate \( b^2 \)
To find \( b^2 \), subtract \( 36 \) from both sides:
\[ b^2 = 81 - 36 \] \[ b^2 = 45 \]
Step 3: Solve for \( b \)
Now we take the square root of both sides to find \( b \):
\[ b = \sqrt{45} \] \[ b = \sqrt{9 \cdot 5} = 3\sqrt{5} \]
Step 4: Check if \( a, b, c \) form a Pythagorean triple
Now we will check if \( a = 6 \), \( b = 3\sqrt{5} \), and \( c = 9 \) can be expressed as integers.
However, a Pythagorean triple consists of three integer values. Since \( \sqrt{5} \) is not an integer, \( b = 3\sqrt{5} \) is not an integer, meaning that \( (a, b, c) = (6, 3\sqrt{5}, 9) \) does not form a Pythagorean triple.
Conclusion
Therefore, the sides \( a = 6 \) and \( c = 9 \) do not form a Pythagorean triple because the resulting side \( b \) is not an integer. Thus, these values do not satisfy the condition for Pythagorean triples.
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