To determine whether a triangle can be constructed with given side lengths, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\):
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
We will apply this theorem to each set of side lengths.
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Side lengths of 5 ft, 12 ft, and 13 ft:
- \(5 + 12 = 17 > 13\)
- \(5 + 13 = 18 > 12\)
- \(12 + 13 = 25 > 5\)
- All conditions are satisfied; therefore, this triangle can be constructed.
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Side lengths of 2 ft, 11 ft, and 15 ft:
- \(2 + 11 = 13 \not> 15\)
- One of the conditions fails; therefore, this triangle cannot be constructed.
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Side lengths of 3 ft, 7 ft, and 11 ft:
- \(3 + 7 = 10 \not> 11\)
- One of the conditions fails; therefore, this triangle cannot be constructed.
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Side lengths of 4 ft, 8 ft, and 15 ft:
- \(4 + 8 = 12 \not> 15\)
- One of the conditions fails; therefore, this triangle cannot be constructed.
In conclusion, the only triangle that can be constructed is the one with side lengths of 5 ft, 12 ft, and 13 ft.