Question

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

20, 10, 30
20, 10, 30

10, 20, 15
10, 20, 15

8, 8, 20
8, 8, 20

8, 4, 12

Answer 1

To determine whether three side lengths can form a triangle, we can apply the Triangle Inequality Theorem, which states that for any triangle with sides \(a\), \(b\), and \(c\), the following inequalities must hold:

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

Let's evaluate each set of side lengths:

1. **20, 10, 30**
- \(20 + 10 = 30\) (not greater)
- \(20 + 30 = 50\) (greater)
- \(10 + 30 = 40\) (greater)
- Does not satisfy the first condition. **Cannot form a triangle.**

2. **10, 20, 15**
- \(10 + 20 = 30\) (greater)
- \(10 + 15 = 25\) (greater)
- \(20 + 15 = 35\) (greater)
- All conditions satisfied. **Forms a triangle.**

3. **8, 8, 20**
- \(8 + 8 = 16\) (not greater)
- \(8 + 20 = 28\) (greater)
- \(8 + 20 = 28\) (greater)
- Does not satisfy the first condition. **Cannot form a triangle.**

4. **8, 4, 12**
- \(8 + 4 = 12\) (not greater)
- \(8 + 12 = 20\) (greater)
- \(4 + 12 = 16\) (greater)
- Does not satisfy the first condition. **Cannot form a triangle.**

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