Question

E) Sarah has toothpicks that are 4 different lengths, 3 in, 8in, 10in, 11in. She is trying to make as many
triangles as she can. How many of the combinations below will make triangles according to the triangle
inequality theorem? Pick ALL that apply (3 points)
3,10,11
8,10,11
3,8,10
3,8,11

Answer 1

To determine which combinations of toothpick lengths can form triangles, we will use the Triangle Inequality Theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold:

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

We will analyze each combination of lengths.

1. **Combination: 3, 10, 11**
- \(3 + 10 = 13 > 11\) (True)
- \(3 + 11 = 14 > 10\) (True)
- \(10 + 11 = 21 > 3\) (True)
- **Result: This combination can form a triangle.**

2. **Combination: 8, 10, 11**
- \(8 + 10 = 18 > 11\) (True)
- \(8 + 11 = 19 > 10\) (True)
- \(10 + 11 = 21 > 8\) (True)
- **Result: This combination can form a triangle.**

3. **Combination: 3, 8, 10**
- \(3 + 8 = 11 > 10\) (True)
- \(3 + 10 = 13 > 8\) (True)
- \(8 + 10 = 18 > 3\) (True)
- **Result: This combination can form a triangle.**

4. **Combination: 3, 8, 11**
- \(3 + 8 = 11 \not> 11\) (False)
- **Result: This combination cannot form a triangle.**

### Summary of Combinations:
- **3, 10, 11**: Can form a triangle.
- **8, 10, 11**: Can form a triangle.
- **3, 8, 10**: Can form a triangle.
- **3, 8, 11**: Cannot form a triangle.

### Final Answer:
The combinations that can form a triangle are **3, 10, 11**; **8, 10, 11**; and **3, 8, 10**.