To determine which set of side lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\):
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
We will apply this theorem to each set of lengths provided.
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5, 9, 14:
- \(5 + 9 = 14\) (not greater)
- \(5 + 14 = 19 > 9\)
- \(9 + 14 = 23 > 5\)
- Conclusion: This set cannot form a triangle.
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510, 15:
- This set does not have three lengths (only two provided).
- Conclusion: This set does not represent a triangle.
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8, 10, 16:
- \(8 + 10 = 18 > 16\)
- \(8 + 16 = 24 > 10\)
- \(10 + 16 = 26 > 8\)
- Conclusion: This set can form a triangle.
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3, 15, 17:
- \(3 + 15 = 18 > 17\)
- \(3 + 17 = 20 > 15\)
- \(15 + 17 = 32 > 3\)
- Conclusion: This set can form a triangle.
Final Results: The side lengths that can form a triangle are:
- 8, 10, 16
- 3, 15, 17
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