Question

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)
Responses

3,8,10
3,8,10

3,8,11
3,8,11

3,10,11
3,10,11

8,10,11

Answer 1

To determine which combinations can create a triangle according to the triangle inequality theorem, we need to check each combination of toothpick lengths to see if they satisfy the following conditions:

For any three lengths \( a, b, c \):
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

We will verify each of the combinations listed:

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

2. **Combination: 3, 8, 11**
- Check: \( 3 + 8 > 11 \) → \( 11 > 11 \) (False)
- Check: \( 3 + 11 > 8 \) → \( 14 > 8 \) (True)
- Check: \( 8 + 11 > 3 \) → \( 19 > 3 \) (True)
- **This combination cannot form a triangle.**

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

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

### Summary of Results:
- **Combinations that can form triangles**:
- **3, 8, 10**
- **3, 10, 11**
- **8, 10, 11**

So the valid combinations are:
- **3, 8, 10**
- **3, 10, 11**
- **8, 10, 11**