Question

Joanna has four straws of different lengths. The table shows the lengths of the straws. Straw A B C D Length in inches 7 9 3 10 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 your work

Answer 1

To determine how many different triangles Joanna can make with the given straws, we need to check which combinations of three straws satisfy the triangle inequality theorem. This theorem states that for any three sides (let's say \(a\), \(b\), and \(c\)), the following must hold true:

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

The lengths of the straws are as follows:
- Straw A: 7 inches
- Straw B: 9 inches
- Straw C: 3 inches
- Straw D: 10 inches

First, we can list all possible combinations of three straws from the four available straws. The combinations of three straws can be determined as follows:

1. A, B, C (7, 9, 3)
2. A, B, D (7, 9, 10)
3. A, C, D (7, 3, 10)
4. B, C, D (9, 3, 10)

Now, we'll check each combination against the triangle inequality theorem:

### Combination 1: A, B, C (7, 9, 3)
- \(7 + 9 > 3\) → 16 > 3 (True)
- \(7 + 3 > 9\) → 10 > 9 (True)
- \(9 + 3 > 7\) → 12 > 7 (True)

All inequalities are satisfied. **This combination can form a triangle.**

### Combination 2: A, B, D (7, 9, 10)
- \(7 + 9 > 10\) → 16 > 10 (True)
- \(7 + 10 > 9\) → 17 > 9 (True)
- \(9 + 10 > 7\) → 19 > 7 (True)

All inequalities are satisfied. **This combination can form a triangle.**

### Combination 3: A, C, D (7, 3, 10)
- \(7 + 3 > 10\) → 10 > 10 (False)
- \(7 + 10 > 3\) → 17 > 3 (True)
- \(3 + 10 > 7\) → 13 > 7 (True)

The first inequality is not satisfied. **This combination cannot form a triangle.**

### Combination 4: B, C, D (9, 3, 10)
- \(9 + 3 > 10\) → 12 > 10 (True)
- \(9 + 10 > 3\) → 19 > 3 (True)
- \(3 + 10 > 9\) → 13 > 9 (True)

All inequalities are satisfied. **This combination can form a triangle.**

### Summary
The valid combinations that can form triangles are:
1. A, B, C (7, 9, 3)
2. A, B, D (7, 9, 10)
3. B, C, D (9, 3, 10)

Thus, Joanna can make a total of **3 different triangles** with the straws:
- Triangle from (7, 9, 3)
- Triangle from (7, 9, 10)
- Triangle from (9, 3, 10)

Therefore, the answer is **3 different triangles** can be made.