To determine which set of three side lengths can form a triangle, we will apply the Triangle Inequality Theorem. The theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
We will check each of the given options:
A. 8, 8, 20
- \(8 + 8 > 20\) → \(16 > 20\) (False)
- \(8 + 20 > 8\) → \(28 > 8\) (True)
- \(8 + 20 > 8\) → \(28 > 8\) (True)
Since the first condition is false, these lengths cannot form a triangle.
B. 20, 10, 30
- \(20 + 10 > 30\) → \(30 > 30\) (False)
- \(20 + 30 > 10\) → \(50 > 10\) (True)
- \(10 + 30 > 20\) → \(40 > 20\) (True)
Since the first condition is false, these lengths cannot form a triangle.
C. 8, 4, 12
- \(8 + 4 > 12\) → \(12 > 12\) (False)
- \(8 + 12 > 4\) → \(20 > 4\) (True)
- \(4 + 12 > 8\) → \(16 > 8\) (True)
Since the first condition is false, these lengths cannot form a triangle.
D. 10, 20, 15
- \(10 + 20 > 15\) → \(30 > 15\) (True)
- \(10 + 15 > 20\) → \(25 > 20\) (True)
- \(20 + 15 > 10\) → \(35 > 10\) (True)
All conditions are satisfied, so these lengths can form a triangle.
Conclusion: The only set of side lengths that can form a triangle is D. 10, 20, 15.
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