To determine if three side lengths can form a triangle, we can use the Triangle Inequality Theorem. According to this theorem, for any three lengths \(a\), \(b\), and \(c\), the following inequalities must all hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's evaluate the given sets of side lengths:
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8, 8, 20:
- \(8 + 8 > 20 \quad \text{(False)}\)
- \(8 + 20 > 8 \quad \text{(True)}\)
- \(8 + 20 > 8 \quad \text{(True)}\)
Since one of the inequalities is false, these lengths cannot form a triangle.
-
20, 10, 30:
- \(20 + 10 > 30 \quad \text{(False)}\)
- \(20 + 30 > 10 \quad \text{(True)}\)
- \(10 + 30 > 20 \quad \text{(True)}\)
Since one of the inequalities is false, these lengths cannot form a triangle.
-
8, 4, 12:
- \(8 + 4 > 12 \quad \text{(False)}\)
- \(8 + 12 > 4 \quad \text{(True)}\)
- \(4 + 12 > 8 \quad \text{(True)}\)
Since one of the inequalities is false, these lengths cannot form a triangle.
-
10, 20, 15:
- \(10 + 20 > 15 \quad \text{(True)}\)
- \(10 + 15 > 20 \quad \text{(False)}\)
- \(20 + 15 > 10 \quad \text{(True)}\)
Since one of the inequalities is false, these lengths cannot form a triangle.
From the evaluations, none of the provided sets of lengths can form a triangle.
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