Question

Apply the Triangle Inequality Theorem to determine which three side lengths form a triangle.(1 point)
Responses

10, 20, 15
10, 20, 15

8, 8, 20
8, 8, 20

8, 4, 12
8, 4, 12

20, 10, 30
20, 10, 30

Answer 1

The Triangle Inequality Theorem states that for any three lengths \( a \), \( b \), and \( c \) to form a triangle, the following conditions must be satisfied:

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

Let's check each set of side lengths:

1. **10, 20, 15**
- \( 10 + 20 = 30 > 15 \) (True)
- \( 10 + 15 = 25 > 20 \) (True)
- \( 20 + 15 = 35 > 10 \) (True)
- **These can form a triangle.**

2. **8, 8, 20**
- \( 8 + 8 = 16 \not> 20 \) (False)
- **These cannot form a triangle.**

3. **8, 4, 12**
- \( 8 + 4 = 12 \not> 12 \) (False)
- **These cannot form a triangle.**

4. **20, 10, 30**
- \( 20 + 10 = 30 \not> 30 \) (False)
- **These cannot form a triangle.**

Based on the evaluations, the only set of lengths that can form a triangle is **10, 20, 15**.