Question

Joanna has four straws of different lengths. The table shows the lengths of the straws.Straw A𝐴 B𝐵 C𝐶 D𝐷 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?

Answer 1

To determine how many different triangles Joanna can make using her straws, we need to apply the triangle inequality theorem. The triangle inequality states that for any three lengths \( a \), \( b \), and \( c \) to form a triangle, the following conditions must be true:

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

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

Now we can examine the combinations of three straws from the four available straws (A, B, C, D).

The combinations of three straws are:
1. A, B, C
2. A, B, D
3. A, C, D
4. B, C, D

Now we'll check each combination to see if it can form a triangle.

1. **Combination A, B, C (7, 9, 3)**:
\[
7 + 9 = 16 > 3 \quad (valid) \\
7 + 3 = 10 \not> 9 \quad (not valid) \\
9 + 3 = 12 > 7 \quad (valid)
\]
This combination **cannot** form a triangle.

2. **Combination A, B, D (7, 9, 10)**:
\[
7 + 9 = 16 > 10 \quad (valid) \\
7 + 10 = 17 > 9 \quad (valid) \\
9 + 10 = 19 > 7 \quad (valid)
\]
This combination **can** form a triangle.

3. **Combination A, C, D (7, 3, 10)**:
\[
7 + 3 = 10 \not> 10 \quad (not valid) \\
7 + 10 = 17 > 3 \quad (valid) \\
3 + 10 = 13 > 7 \quad (valid)
\]
This combination **cannot** form a triangle.

4. **Combination B, C, D (9, 3, 10)**:
\[
9 + 3 = 12 > 10 \quad (valid) \\
9 + 10 = 19 > 3 \quad (valid) \\
3 + 10 = 13 > 9 \quad (valid)
\]
This combination **can** form a triangle.

Based on the analysis above:
- The combinations that can form a triangle are **A, B, D (7, 9, 10)** and **B, C, D (9, 3, 10)**.
- The total number of different triangles Joanna can make is **2**.

### Summary:
**Number of different triangles that can be formed:** 2
**Combinations of straws used:**
1. A, B, D (7, 9, 10)
2. B, C, D (9, 3, 10)