Question

Is (5, 7, 74−−√ ) a Pythagorean triple? (1 point) Responses Yes, it is a Pythagorean triple. Yes, it is a Pythagorean triple. No, it is not a Pythagorean triple. No, it is not a Pythagorean triple. It cannot be determined.

Answer 1

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.

1. 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:
\[
a^2 + b^2 = 25 + 49 = 74
\]

2. Calculate \(c^2\):
- \(c = \sqrt{74}\), so \(c^2 = (\sqrt{74})^2 = 74\)

3. 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.**