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 straws, we can apply the triangle inequality theorem. According to the triangle inequality, for three lengths \(a\), \(b\), and \(c\) to form a triangle, the following three conditions must be met:

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

Where \(a\), \(b\), and \(c\) are the lengths of the straws used.

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

First, we will identify all the combinations of three straws from the four available straws (A, B, C, D). The combinations 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 to see if they can form a triangle.

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

Since all conditions are satisfied, this combination can form a triangle.

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

Since all conditions are satisfied, this combination can form a triangle.

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

Since one condition is not satisfied, this combination cannot form a triangle.

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

Since all conditions are satisfied, this combination can form a triangle.

### Summary of Results:
Joanna can make triangles using the following combinations:
1. A, B, C (7, 9, 3)
2. A, B, D (7, 9, 10)
3. B, C, D (9, 3, 10)

Thus, Joanna can form a total of **3 different triangles**.