To determine which set of side lengths does not form a triangle, we can use the Triangle Inequality Theorem. This theorem states that for any three sides \(a\), \(b\), and \(c\), the following must hold true:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's apply this theorem to each set of side lengths:
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For 2, 3, 5:
- \(2 + 3 = 5\) (not greater, so this does not satisfy the triangle inequality)
- \(2 + 5 = 7\) (greater)
- \(3 + 5 = 8\) (greater)
- At least one condition fails, so these lengths do not form a triangle.
-
For 3, 5, 7:
- \(3 + 5 = 8\) (greater)
- \(3 + 7 = 10\) (greater)
- \(5 + 7 = 12\) (greater)
- All conditions satisfied, so these lengths do form a triangle.
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For 5, 7, 10:
- \(5 + 7 = 12\) (greater)
- \(5 + 10 = 15\) (greater)
- \(7 + 10 = 17\) (greater)
- All conditions satisfied, so these lengths do form a triangle.
-
For 6, 14, 18:
- \(6 + 14 = 20\) (greater)
- \(6 + 18 = 24\) (greater)
- \(14 + 18 = 32\) (greater)
- All conditions satisfied, so these lengths do form a triangle.
The only set of lengths that does not form a triangle is 2, 3, 5.