To determine how many different triangles Joanna can make with her straws, we must use the triangle inequality theorem, which states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following must hold true:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.
In this case, Joanna has the following straws:
- Straw A: 7 inches
- Straw B: 9 inches
- Straw C: 3 inches
- Straw D: 10 inches
We will consider all combinations of three straws and check if they satisfy the triangle inequality theorem.
The combinations of three straws from the four available are:
- A, B, C (7, 9, 3)
- A, B, D (7, 9, 10)
- A, C, D (7, 3, 10)
- B, C, D (9, 3, 10)
Now, let's check each combination:
Combination 1: A, B, C (7, 9, 3)
- Check \(7 + 9 > 3\): 16 > 3 (True)
- Check \(7 + 3 > 9\): 10 > 9 (True)
- Check \(9 + 3 > 7\): 12 > 7 (True)
This combination is valid.
Combination 2: A, B, D (7, 9, 10)
- Check \(7 + 9 > 10\): 16 > 10 (True)
- Check \(7 + 10 > 9\): 17 > 9 (True)
- Check \(9 + 10 > 7\): 19 > 7 (True)
This combination is valid.
Combination 3: A, C, D (7, 3, 10)
- Check \(7 + 3 > 10\): 10 > 10 (False)
- Check \(7 + 10 > 3\): 17 > 3 (True)
- Check \(3 + 10 > 7\): 13 > 7 (True)
This combination is not valid because \(7 + 3 \leq 10\).
Combination 4: B, C, D (9, 3, 10)
- Check \(9 + 3 > 10\): 12 > 10 (True)
- Check \(9 + 10 > 3\): 19 > 3 (True)
- Check \(3 + 10 > 9\): 13 > 9 (True)
This combination is valid.
Summary of valid combinations:
- A, B, C (7, 9, 3)
- A, B, D (7, 9, 10)
- B, C, D (9, 3, 10)
Thus, Joanna can make 3 different triangles using the valid combinations of straws.
Answer:
Joanna can make 3 different triangles using the following combinations of straws:
- A (7), B (9), C (3)
- A (7), B (9), D (10)
- B (9), C (3), D (10)
1 answer