Question

Joanna has four straws of different lengths. The table shows the lengths of the straws.Straw A𝐴 B𝐵 C𝐶 D𝐷 Length in inches 7 9 3 10Joanna 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.(4 points)

Answer 1

To determine how many different triangles Joanna can make with her straws, we must use the triangle inequality theorem, which states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following must hold true:

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

Where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.

In this case, Joanna has the following straws:

- Straw A: 7 inches
- Straw B: 9 inches
- Straw C: 3 inches
- Straw D: 10 inches

We will consider all combinations of three straws and check if they satisfy the triangle inequality theorem.

The combinations of three straws from the four available are:

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, let's check each combination:

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

This combination is valid.

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

This combination is valid.

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

This combination is **not valid** because \(7 + 3 \leq 10\).

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

This combination is valid.

### Summary of valid combinations:
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 3 different triangles using the valid combinations of straws.

### Answer:
Joanna can make **3 different triangles** using the following combinations of straws:
- A (7), B (9), C (3)
- A (7), B (9), D (10)
- B (9), C (3), D (10)