Question

3-4 sentences Joanna has four straws of different lengths. The table shows the lengths of the straws.%0D%0A%0D%0ALength of Straws%0D%0AStraw%09A%0D%0A𝐴%0D%0AB%0D%0A𝐵%0D%0AC%0D%0A𝐶%0D%0AD%0D%0A𝐷%0D%0ALength in inches%097%099%093%0910%0D%0AJoanna 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 using her four straws of lengths 3 inches, 7 inches, 9 inches, and 10 inches, we must consider the combinations of straws that can satisfy the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must hold true:

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

Let's label the straws as follows:

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

Now, we will test all combinations of three straws:

1. **Combination (A, B, C)**:
- Lengths: 3, 7, 9
- Check:
- \(3 + 7 = 10\) (not greater than 9)
- No triangle can be formed.

2. **Combination (A, B, D)**:
- Lengths: 3, 7, 10
- Check:
- \(3 + 7 = 10\) (not greater than 10)
- No triangle can be formed.

3. **Combination (A, C, D)**:
- Lengths: 3, 9, 10
- Check:
- \(3 + 9 = 12 > 10\)
- \(3 + 10 = 13 > 9\)
- \(9 + 10 = 19 > 3\)
- A triangle can be formed.

4. **Combination (B, C, D)**:
- Lengths: 7, 9, 10
- Check:
- \(7 + 9 = 16 > 10\)
- \(7 + 10 = 17 > 9\)
- \(9 + 10 = 19 > 7\)
- A triangle can be formed.

Thus, out of all combinations, Joanna can form triangles using the pairs:
- Combination (A, C, D): 3, 9, 10 inches
- Combination (B, C, D): 7, 9, 10 inches

In total, Joanna can create **2 different triangles**.