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Apply the Triangle Inequality Theorem to determine which three side lengths form a triangle. Responses:
A. 8, 8, 20,
B. 20, 10, 30
C. 8, 4, 12
D. 10, 20, 15

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Answered by GPT-4o mini
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:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

We will check each of the given options:

**A. 8, 8, 20**

1. \(8 + 8 > 20\) → \(16 > 20\) (False)
2. \(8 + 20 > 8\) → \(28 > 8\) (True)
3. \(8 + 20 > 8\) → \(28 > 8\) (True)

Since the first condition is false, these lengths cannot form a triangle.

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**B. 20, 10, 30**

1. \(20 + 10 > 30\) → \(30 > 30\) (False)
2. \(20 + 30 > 10\) → \(50 > 10\) (True)
3. \(10 + 30 > 20\) → \(40 > 20\) (True)

Since the first condition is false, these lengths cannot form a triangle.

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**C. 8, 4, 12**

1. \(8 + 4 > 12\) → \(12 > 12\) (False)
2. \(8 + 12 > 4\) → \(20 > 4\) (True)
3. \(4 + 12 > 8\) → \(16 > 8\) (True)

Since the first condition is false, these lengths cannot form a triangle.

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**D. 10, 20, 15**

1. \(10 + 20 > 15\) → \(30 > 15\) (True)
2. \(10 + 15 > 20\) → \(25 > 20\) (True)
3. \(20 + 15 > 10\) → \(35 > 10\) (True)

All conditions are satisfied, so these lengths can form a triangle.

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**Conclusion:** The only set of side lengths that can form a triangle is **D. 10, 20, 15**.

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