Question

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

Straw

ABCD

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

Answer 1

To determine how many different triangles can be formed using three of Joanna's straws, we must use the triangle inequality theorem, which states that for any three lengths \( a \), \( b \), and \( c \) (where \( a \leq b \leq c \)), the following conditions must hold:

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

In the case of straws, since all straws have positive lengths, the first condition \( a + b > c \) is sufficient to verify that three lengths can form a triangle because the other two conditions will be satisfied if the first is.

The lengths of the straws are:
- \( A = 7 \) inches
- \( B = 9 \) inches
- \( C = 3 \) inches
- \( D = 10 \) inches

Now, let's list all combinations of three straws and check if they can form a triangle.

1. **Combination: 7, 9, 3**
- Check: \( 3 + 7 > 9 \) (False)
- Cannot form a triangle.

2. **Combination: 7, 9, 10**
- Check: \( 7 + 9 > 10 \) (True)
- Can form a triangle.

3. **Combination: 7, 3, 10**
- Check: \( 3 + 7 > 10 \) (False)
- Cannot form a triangle.

4. **Combination: 9, 3, 10**
- Check: \( 3 + 9 > 10 \) (False)
- Cannot form a triangle.

From our checks, we see that the only combination that can form a triangle is the combination of the 7, 9, and 10 inch straws.

So, Joanna can make **1 different triangle** using the following combination of straws:
- **7 inches, 9 inches, 10 inches**.