To determine if the triangle is a Pythagorean triple, we need to check whether the lengths of the sides satisfy the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \):
\[ a^2 + b^2 = c^2 \]
In this case, we know:
- One leg \( A = 7 \) units
- Hypotenuse \( C = 9 \) units
We need to find the length of the other leg, which we will call \( B \).
Step 1: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ A^2 + B^2 = C^2 \]
Substituting the known values:
\[ 7^2 + B^2 = 9^2 \]
Calculating \( 7^2 \) and \( 9^2 \):
\[ 49 + B^2 = 81 \]
Step 2: Solve for \( B^2 \)
Now, subtract \( 49 \) from both sides:
\[ B^2 = 81 - 49 \] \[ B^2 = 32 \]
Step 3: Calculate \( B \)
To find \( B \), we take the square root:
\[ B = \sqrt{32} = 4\sqrt{2} \]
Step 4: Check if the sides form a Pythagorean triple
A Pythagorean triple consists of three integers. Here, we have the sides as:
- \( A = 7 \) (integer)
- \( B = 4\sqrt{2} \) (not an integer)
- \( C = 9 \) (integer)
Since \( B \) is not an integer, the sides \( 7, 4\sqrt{2}, 9 \) do not form a Pythagorean triple.
Conclusion
A. Is it a Pythagorean triple? No, it is not a Pythagorean triple.
Explanation: For a set of three lengths to form a Pythagorean triple, all three lengths must be integers. In this case, while \( A \) and \( C \) are integers, \( B \) is not, as it equals \( 4\sqrt{2} \), which is an irrational number. Therefore, the triangle does not satisfy the criteria for being a Pythagorean triple.
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