Question

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

Straw A B C 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.

Answer 1

To determine how many different triangles Joanna can make using combinations of three of the four straws, we need to evaluate the lengths of each combination to confirm whether they can form a triangle. According to the triangle inequality theorem, for three lengths \( a \), \( b \), and \( c \) to form a triangle, the following conditions must hold true:

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

Given the lengths of the straws:
- Straw A: 7 inches
- Straw B: 9 inches
- Straw C: 3 inches
- Straw D: 10 inches

We will examine all combinations of three straws:

### Step 1: List the combinations of three straws
1. A, B, C (7, 9, 3)
2. A, B, D (7, 9, 10)
3. A, C, D (7, 3, 10)
4. B, C, D (9, 3, 10)

### Step 2: Check each combination to see if it forms a triangle

#### Combination 1: A, B, C (7, 9, 3)
- \( 7 + 9 > 3 \) → \( 16 > 3 \) (True)
- \( 7 + 3 > 9 \) → \( 10 > 9 \) (True)
- \( 9 + 3 > 7 \) → \( 12 > 7 \) (True)

**Conclusion**: This combination forms a triangle.

#### Combination 2: A, B, D (7, 9, 10)
- \( 7 + 9 > 10 \) → \( 16 > 10 \) (True)
- \( 7 + 10 > 9 \) → \( 17 > 9 \) (True)
- \( 9 + 10 > 7 \) → \( 19 > 7 \) (True)

**Conclusion**: This combination forms a triangle.

#### Combination 3: A, C, D (7, 3, 10)
- \( 7 + 3 > 10 \) → \( 10 > 10 \) (False)
- \( 7 + 10 > 3 \) → \( 17 > 3 \) (True)
- \( 3 + 10 > 7 \) → \( 13 > 7 \) (True)

**Conclusion**: This combination does not form a triangle.

#### Combination 4: B, C, D (9, 3, 10)
- \( 9 + 3 > 10 \) → \( 12 > 10 \) (True)
- \( 9 + 10 > 3 \) → \( 19 > 3 \) (True)
- \( 3 + 10 > 9 \) → \( 13 > 9 \) (True)

**Conclusion**: This combination forms a triangle.

### Step 3: Final Count and Combinations
The combinations that form triangles are:
1. A, B, C (7, 9, 3)
2. A, B, D (7, 9, 10)
3. B, C, D (9, 3, 10)

Thus, Joanna can make **three different triangles** using the following combinations:
1. 7, 9, 3
2. 7, 9, 10
3. 9, 3, 10

**Final Answer**: Joanna can make **3 different triangles** using the straws.