To determine whether the given lengths can be sides of a right triangle, we can use the Pythagorean theorem, which states that for a triangle with sides \(a\), \(b\), and hypotenuse \(c\), the following equation must be true:
\[ a^2 + b^2 = c^2 \]
where \(c\) is the longest side.
Let's analyze each set of lengths provided.
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For lengths 14, 24, and 26:
- The longest side is 26. We check if: \[ 14^2 + 24^2 = 26^2 \]
- Calculate \(14^2\), \(24^2\), and \(26^2\):
- \(14^2 = 196\)
- \(24^2 = 576\)
- \(26^2 = 676\)
- Now, check if: \[ 196 + 576 = 676 \]
- \(772 \neq 676\). Thus, lengths 14, 24, and 26 cannot form a right triangle.
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For lengths 30, 72, and 78:
- The longest side is 78. We check if: \[ 30^2 + 72^2 = 78^2 \]
- Calculate \(30^2\), \(72^2\), and \(78^2\):
- \(30^2 = 900\)
- \(72^2 = 5184\)
- \(78^2 = 6084\)
- Now check if: \[ 900 + 5184 = 6084 \]
- \(6084 = 6084\). Thus, lengths 30, 72, and 78 can form a right triangle.
Now we summarize the findings:
- The lengths 14, 24, and 26 cannot be sides of a right triangle.
- The lengths 30, 72, and 78 can be sides of a right triangle.
Thus, the correct statement is: The lengths 14, 24 and 26 can not be sides of a right triangle. The lengths 30, 72, and 78 can be sides of a right triangle.