Question

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

Length of Straws
Straw A
𝐴
B
𝐵
C
𝐶
D
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.(4 points)

Answer 1

To determine how many different triangles Joanna can make using combinations of three straws from the lengths given (7 inches, 9 inches, 3 inches, and 10 inches), we need to use the triangle inequality theorem. According to this theorem, for three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Let's denote the lengths of the straws:
- Straw A = 7 inches
- Straw B = 9 inches
- Straw C = 3 inches
- Straw D = 10 inches

We will evaluate all possible combinations of three straws to see if they can form a triangle.

### Possible combinations:
1. **Straws A, B, C (7, 9, 3)**
- Check the triangle inequalities:
- 7 + 9 > 3 (True)
- 7 + 3 > 9 (False)
- 9 + 3 > 7 (True)
- **Cannot form a triangle.**

2. **Straws A, B, D (7, 9, 10)**
- Check the triangle inequalities:
- 7 + 9 > 10 (True)
- 7 + 10 > 9 (True)
- 9 + 10 > 7 (True)
- **Can form a triangle.**

3. **Straws A, C, D (7, 3, 10)**
- Check the triangle inequalities:
- 7 + 3 > 10 (False)
- 7 + 10 > 3 (True)
- 3 + 10 > 7 (True)
- **Cannot form a triangle.**

4. **Straws B, C, D (9, 3, 10)**
- Check the triangle inequalities:
- 9 + 3 > 10 (False)
- 9 + 10 > 3 (True)
- 3 + 10 > 9 (True)
- **Cannot form a triangle.**

### Conclusion:
Joanna can only form triangles with the combination of **Straws A, B, and D (7, 9, 10)**.

Thus, there is a total of **1 different triangle** that she can make.

**Final answer: 1 triangle (using straws A, B, D).**