Question

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

8, 3, 12
8, 3, 12

10, 9, 1
10, 9, 1

1, 2, 3
1, 2, 3

8, 10, 3

Answer 1

To determine if a set of side lengths can form a triangle, we can use the Triangle Inequality Theorem. This theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), the following inequalities must hold:

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

Let's apply this to each set of side lengths:

1. **For the lengths 8, 3, 12:**
- \(8 + 3 = 11\) (not greater than 12)
- Therefore, these lengths do not satisfy the triangle inequality; **cannot form a triangle.**

2. **For the lengths 10, 9, 1:**
- \(10 + 9 = 19\) (greater than 1)
- \(10 + 1 = 11\) (greater than 9)
- \(9 + 1 = 10\) (not greater than 10)
- Therefore, these lengths do not satisfy the triangle inequality; **cannot form a triangle.**

3. **For the lengths 1, 2, 3:**
- \(1 + 2 = 3\) (not greater than 3)
- Therefore, these lengths do not satisfy the triangle inequality; **cannot form a triangle.**

4. **For the lengths 8, 10, 3:**
- \(8 + 10 = 18\) (greater than 3)
- \(8 + 3 = 11\) (greater than 10)
- \(10 + 3 = 13\) (greater than 8)
- Therefore, these lengths satisfy all triangle inequalities; **can form a triangle.**

**Summary:**
- **8, 3, 12**: cannot form a triangle
- **10, 9, 1**: cannot form a triangle
- **1, 2, 3**: cannot form a triangle
- **8, 10, 3**: can form a triangle