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.(4 points)

Answer 1

To determine how many different triangles Joanna can make using combinations of three straws, we can apply the triangle inequality theorem. The triangle inequality states that for three sides \(a\), \(b\), and \(c\) (where \(a \leq b \leq c\)), the following must hold true:

\[
a + b > c
\]

First, we list the lengths of the straws:
- Straw A: 7 inches
- Straw B: 9 inches
- Straw C: 3 inches
- Straw D: 10 inches

Next, we will evaluate all possible combinations of three straws from the four available. The combinations of three straws can be formed by taking each possible set from the four straws.

The combinations of straws are:
1. ABC (7, 9, 3)
2. ABD (7, 9, 10)
3. ACD (7, 3, 10)
4. BCD (9, 3, 10)

Now, we will check each combination to see if it satisfies the triangle inequality:

1. **Combination ABC: (7, 9, 3)**
- Sort the lengths: \(3, 7, 9\)
- Check: \(3 + 7 = 10 > 9\) (Yes)
- Check: \(3 + 9 = 12 > 7\) (Yes)
- Check: \(7 + 9 = 16 > 3\) (Yes)
- This combination forms a triangle.

2. **Combination ABD: (7, 9, 10)**
- Sort the lengths: \(7, 9, 10\)
- Check: \(7 + 9 = 16 > 10\) (Yes)
- Check: \(7 + 10 = 17 > 9\) (Yes)
- Check: \(9 + 10 = 19 > 7\) (Yes)
- This combination forms a triangle.

3. **Combination ACD: (7, 3, 10)**
- Sort the lengths: \(3, 7, 10\)
- Check: \(3 + 7 = 10\) (No, it is not greater)
- Since the first condition fails, this combination does NOT form a triangle.

4. **Combination BCD: (9, 3, 10)**
- Sort the lengths: \(3, 9, 10\)
- Check: \(3 + 9 = 12 > 10\) (Yes)
- Check: \(3 + 10 = 13 > 9\) (Yes)
- Check: \(9 + 10 = 19 > 3\) (Yes)
- This combination forms a triangle.

In summary, the combinations that form valid triangles are:

- ABC (7, 9, 3)
- ABD (7, 9, 10)
- BCD (9, 3, 10)

The total number of different triangles that Joanna can make is **3**.

**Final Answer:**
Joanna can form 3 different triangles with the following combinations:
1. ABC (7, 9, 3)
2. ABD (7, 9, 10)
3. BCD (9, 3, 10)