Question

Joanna has four straws of different lengths. The table shows the lengths of the straws.



Joanna makes as many triangles as she can using combinations of three of these straws. Determine how many different triangles she can make. What combinations of straws does she use?

Show ALL of your work for credit. (Either type your work in the text box below or upload an image.)

(4 points)

Answer 1

To determine how many different triangles Joanna can make using combinations of three straws, we need to apply the triangle inequality theorem. This theorem states that for any three lengths \( a \), \( b \), and \( c \) (where \( a \leq b \leq c \)), they can form a triangle if and only if \( a + b > c \).

Let’s denote the lengths of the straws, which I'll assume are \( a_1, a_2, a_3, a_4 \). For illustration purposes, let's assume the lengths are:

- Straw 1: 3
- Straw 2: 4
- Straw 3: 5
- Straw 4: 6

### Step 1: List all combinations

Given these four straws, we will list all the combinations of three straws:

1. \( 3, 4, 5 \)
2. \( 3, 4, 6 \)
3. \( 3, 5, 6 \)
4. \( 4, 5, 6 \)

### Step 2: Apply the triangle inequality

Now, we will check each combination to see if they satisfy the triangle inequality.

1. **Combination \( 3, 4, 5 \)**:
- Check \( 3 + 4 > 5 \): \( 7 > 5 \) (True)
- Check \( 3 + 5 > 4 \): \( 8 > 4 \) (True)
- Check \( 4 + 5 > 3 \): \( 9 > 3 \) (True)
- **This combination can form a triangle.**

2. **Combination \( 3, 4, 6 \)**:
- Check \( 3 + 4 > 6 \): \( 7 > 6 \) (True)
- Check \( 3 + 6 > 4 \): \( 9 > 4 \) (True)
- Check \( 4 + 6 > 3 \): \( 10 > 3 \) (True)
- **This combination can form a triangle.**

3. **Combination \( 3, 5, 6 \)**:
- Check \( 3 + 5 > 6 \): \( 8 > 6 \) (True)
- Check \( 3 + 6 > 5 \): \( 9 > 5 \) (True)
- Check \( 5 + 6 > 3 \): \( 11 > 3 \) (True)
- **This combination can form a triangle.**

4. **Combination \( 4, 5, 6 \)**:
- Check \( 4 + 5 > 6 \): \( 9 > 6 \) (True)
- Check \( 4 + 6 > 5 \): \( 10 > 5 \) (True)
- Check \( 5 + 6 > 4 \): \( 11 > 4 \) (True)
- **This combination can form a triangle.**

### Step 3: Conclusion

All combinations of straws (3, 4, 5), (3, 4, 6), (3, 5, 6), and (4, 5, 6) can form a triangle.

Thus, the total number of different triangles that Joanna can make is **4**.

**Summary of combinations that can form triangles:**
- \( 3, 4, 5 \)
- \( 3, 4, 6 \)
- \( 3, 5, 6 \)
- \( 4, 5, 6 \)

Total triangles: **4**.