No, the sides of a triangle cannot have lengths 1, 1, and 4.
To determine whether three lengths can form a triangle, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
Let's apply this theorem to the sides 1, 1, and 4:
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Check if \(1 + 1 > 4\): \[ 2 > 4 \quad \text{(False)} \]
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Check if \(1 + 4 > 1\): \[ 5 > 1 \quad \text{(True)} \]
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Check if \(1 + 4 > 1\): \[ 5 > 1 \quad \text{(True)} \]
Since the first condition is false, the side lengths 1, 1, and 4 cannot form a triangle. Therefore, the answer is no.