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 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 using combinations of three straws, we need to consider the triangle inequality theorem. The theorem states that for any three lengths \(a\), \(b\), and \(c\) (where \(a \leq b \leq c\)), a triangle can be formed if:

\[
a + b > c
\]

Joanna has four straws of lengths:

- \(A = 7\) inches
- \(B = 9\) inches
- \(C = 3\) inches
- \(D = 10\) inches

We can find all combinations of three straws and check if they satisfy the triangle inequality.

The combinations of three straws are:

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)

Now we will check each combination against the triangle inequality.

### 1. Combination \(A, B, C\) (7, 9, 3)
- Ordered: \(3, 7, 9\)
- Check: \(3 + 7 = 10 > 9\) ✅. Therefore, this combination can form a triangle.

### 2. Combination \(A, B, D\) (7, 9, 10)
- Ordered: \(7, 9, 10\)
- Check: \(7 + 9 = 16 > 10\) ✅. Therefore, this combination can form a triangle.

### 3. Combination \(A, C, D\) (7, 3, 10)
- Ordered: \(3, 7, 10\)
- Check: \(3 + 7 = 10\) ❌. This combination **cannot** form a triangle since \(3 + 7\) is **not greater** than \(10\).

### 4. Combination \(B, C, D\) (9, 3, 10)
- Ordered: \(3, 9, 10\)
- Check: \(3 + 9 = 12 > 10\) ✅. Therefore, this combination can form a triangle.

### Summary
The valid combinations that can form triangles are:
1. Straw lengths \(A, B, C\) (7, 9, 3)
2. Straw lengths \(A, B, D\) (7, 9, 10)
3. Straw lengths \(B, C, D\) (9, 3, 10)

Thus, Joanna can make **3 different triangles** with the following combinations:
1. \(A, B, C\) (7, 9, 3)
2. \(A, B, D\) (7, 9, 10)
3. \(B, C, D\) (9, 3, 10)

So the final answer is that Joanna can make **3 different triangles**.