Question

Joanna has four straws of different lengths. The table shows the lengths of the straws.StrawABCDLength in inches79310Joanna 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 create with the straws of lengths 7 inches, 9 inches, 3 inches, and 10 inches, we first need to apply the triangle inequality theorem. According to this theorem, for any three lengths \( a \), \( b \), and \( c \) to form a triangle, the following conditions must be satisfied:

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

We will check all combinations of three straws from the four available straws (A, B, C, and D based on their lengths) and see if they meet these conditions.

### Combinations to Consider:
1. **Straws A, B, C**: Lengths 7, 9, 3
2. **Straws A, B, D**: Lengths 7, 9, 10
3. **Straws A, C, D**: Lengths 7, 3, 10
4. **Straws B, C, D**: Lengths 9, 3, 10

We will check each combination to see if they can form a triangle.

### Checking Combinations:

1. **Combination (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)
- **Valid Triangle**

2. **Combination (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)
- **Valid Triangle**

3. **Combination (A, C, D): 7, 3, 10**
- Check \( 7 + 3 > 10 \) → \( 10 > 10 \) (False)
- **Not a valid triangle**

4. **Combination (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)
- **Valid Triangle**

### Summary:
The valid combinations that form triangles are:
- Straws A, B, C (Lengths 7, 9, 3)
- Straws A, B, D (Lengths 7, 9, 10)
- Straws B, C, D (Lengths 9, 3, 10)

### Total Number of Valid Triangles:
Joanna can make **3 different triangles**.

1. **Triangle 1:** Using 7, 9, 3
2. **Triangle 2:** Using 7, 9, 10
3. **Triangle 3:** Using 9, 3, 10

Thus, the final answer is that Joanna can make 3 different triangles and those combinations of straws used are (7, 9, 3), (7, 9, 10), and (9, 3, 10).