To determine whether \((5, 7, \sqrt{74})\) is a Pythagorean triple, we need to check if the equation \(a^2 + b^2 = c^2\) holds, where \(a\) and \(b\) are the two shorter sides, and \(c\) is the hypotenuse.
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Calculate \(a^2\) and \(b^2\):
- \(a = 5\), so \(a^2 = 5^2 = 25\)
- \(b = 7\), so \(b^2 = 7^2 = 49\)
Now, adding these two: \[ a^2 + b^2 = 25 + 49 = 74 \]
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Calculate \(c^2\):
- \(c = \sqrt{74}\), so \(c^2 = (\sqrt{74})^2 = 74\)
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Compare \(a^2 + b^2\) and \(c^2\): \[ a^2 + b^2 = 74 \quad \text{and} \quad c^2 = 74 \] Since \(a^2 + b^2 = c^2\), the condition holds.
Thus, \((5, 7, \sqrt{74})\) is indeed a Pythagorean triple.
So the correct response is: Yes, it is a Pythagorean triple.