Question

Apply the Triangle Inequality Theorem to determine which three side lengths form a triangle.(1 point) Responses 8, 8, 20 8, 8, 20 20, 10, 30 20, 10, 30 8, 4, 12 8, 4, 12 10, 20, 15

Answer 1

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:

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

Let's evaluate the given sets of side lengths:

1. **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.

2. **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.

3. **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.

4. **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.