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 Joanna can make using the four straws of lengths 3, 7, 9, and 10 inches, we need to check which combinations of three straws satisfy the triangle inequality conditions.

The triangle inequality states that for any three sides \(a\), \(b\), and \(c\) (where \(a \leq b \leq c\)), it must hold that:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

However, if we ensure that \(a\), \(b\), and \(c\) are ordered such that \(a \leq b \leq c\), it is sufficient to check just the first condition:
- \(a + b > c\)

Now, we will examine all combinations of three straws from the four lengths given (3, 7, 9, and 10 inches):

### Possible Combinations of Straws
1. **Combination: (3, 7, 9)**
- Check: \(3 + 7 = 10 > 9\) (satisfied)
- Valid triangle.

2. **Combination: (3, 7, 10)**
- Check: \(3 + 7 = 10\) (not satisfied, as it must be greater)
- Not a valid triangle.

3. **Combination: (3, 9, 10)**
- Check: \(3 + 9 = 12 > 10\) (satisfied)
- Valid triangle.

4. **Combination: (7, 9, 10)**
- Check: \(7 + 9 = 16 > 10\) (satisfied)
- Valid triangle.

### Summary of Results
The valid triangles that Joanna can construct from the given straw lengths are:
1. Triangle from (3, 7, 9)
2. Triangle from (3, 9, 10)
3. Triangle from (7, 9, 10)

Therefore, the total number of different triangles Joanna can make is **3**, and the combinations of straws used are:
- (3, 7, 9)
- (3, 9, 10)
- (7, 9, 10)