The triangle inequality theorem states that for any three sides of a triangle with lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Given the sides with lengths 3 feet, 4 feet, and 5 feet, we can label them as follows:
- \(a = 3\)
- \(b = 4\)
- \(c = 5\)
Now, we will check each of the triangle inequalities:
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Check \(a + b > c\): \[ 3 + 4 = 7 > 5 \quad \text{(true)} \]
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Check \(a + c > b\): \[ 3 + 5 = 8 > 4 \quad \text{(true)} \]
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Check \(b + c > a\): \[ 4 + 5 = 9 > 3 \quad \text{(true)} \]
Since all three conditions of the triangle inequality theorem are satisfied, the sides with lengths 3 feet, 4 feet, and 5 feet do indeed form a triangle.
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