To determine if \((5, 7, \sqrt{74})\) is a Pythagorean triple, we need to check if the relationship \(a^2 + b^2 = c^2\) holds, where \(a\) and \(b\) are the two smaller numbers and \(c\) is the largest number.
Here, we have:
- \(a = 5\)
- \(b = 7\)
- \(c = \sqrt{74}\)
Now we calculate \(a^2 + b^2\) and \(c^2\):
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Calculate \(a^2\) and \(b^2\): \[ a^2 = 5^2 = 25 \] \[ b^2 = 7^2 = 49 \]
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Now add these two values: \[ a^2 + b^2 = 25 + 49 = 74 \]
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Calculate \(c^2\): \[ c^2 = (\sqrt{74})^2 = 74 \]
Since \(a^2 + b^2 = 74\) and \(c^2 = 74\), we find that: \[ a^2 + b^2 = c^2 \]
This confirms that \((5, 7, \sqrt{74})\) is indeed a Pythagorean triple.
Therefore, the answer is 1 (for yes).
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