To determine if the given triangle (with one leg \(A = 7\) units and hypotenuse \(C = 9\) units) is a Pythagorean triple, we'll need to follow the Pythagorean theorem.
A. Is it a Pythagorean triple?
A Pythagorean triple consists of three positive integers \(A\), \(B\), and \(C\) that relate to a right triangle according to the formula:
\[ A^2 + B^2 = C^2 \]
where \(C\) is the length of the hypotenuse (the longest side of the triangle).
Steps to Find B and Verify the Triple
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Identify known values:
- Leg \(A = 7\)
- Hypotenuse \(C = 9\)
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Use the Pythagorean theorem to find leg \(B\): Rearranging the Pythagorean theorem gives us:
\[ B^2 = C^2 - A^2 \]
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Calculate \(C^2\) and \(A^2\):
- \(A^2 = 7^2 = 49\)
- \(C^2 = 9^2 = 81\)
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Plug these values into the equation to find \(B^2\):
\[ B^2 = 81 - 49 = 32 \]
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Calculate \(B\): To find \(B\), take the square root of \(B^2\):
\[ B = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \approx 5.66 \]
Conclusion for Part A
Since \(B\) is not an integer (it's approximately 5.66), the triangle with sides \(7\), \(4\sqrt{2}\), and \(9\) does not consist of all integers. Therefore, it is not a Pythagorean triple.
B. Explanation
To summarize why \(7\), \(B\) (which is non-integer), and \(9\) do not form a Pythagorean triple:
- The Pythagorean theorem strictly requires all three values to be integers. In this case, while we found that \(B\) is \(4\sqrt{2}\), which is approximately \(5.66\), it doesn't meet the requirement of being a whole number.
- Pythagorean triples are sets of integers. Common examples include \((3, 4, 5)\), \((5, 12, 13)\), and \((8, 15, 17)\). Since our leg \(B\) is not an integer, the triangle with sides \(7\) and \(9\) does not form a Pythagorean triple.
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