Question

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

Straw A
𝐴
B
𝐵
C
𝐶
D
𝐷

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 the straws of lengths 3 inches (C), 7 inches (A), 9 inches (B), and 10 inches (D), we will apply the triangle inequality theorem. The triangle inequality states that for any three lengths \(a\), \(b\), and \(c\), a triangle can be formed if the following conditions are satisfied:

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

We'll evaluate all possible combinations of three straws from the list:

1. **Combination 1:** (3, 7, 9)
- \(3 + 7 = 10\) (not greater than 9)
- **Cannot form a triangle**

2. **Combination 2:** (3, 7, 10)
- \(3 + 7 = 10\) (not greater than 10)
- **Cannot form a triangle**

3. **Combination 3:** (3, 9, 10)
- \(3 + 9 = 12 > 10\)
- \(3 + 10 = 13 > 9\)
- \(9 + 10 = 19 > 3\)
- **Can form a triangle**

4. **Combination 4:** (7, 9, 10)
- \(7 + 9 = 16 > 10\)
- \(7 + 10 = 17 > 9\)
- \(9 + 10 = 19 > 7\)
- **Can form a triangle**

After analyzing all combinations of three lengths, the only combinations that can form triangles are:

- (3, 9, 10)
- (7, 9, 10)

Therefore, Joanna can make a total of **2 different triangles** using the straws of lengths 3, 9, and 10 inches, and 7, 9, and 10 inches.