To determine how many different triangles Joanna can make using three of the four straws of different lengths, we can use the triangle inequality theorem. According to this theorem, for three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Given the lengths of the straws:
- A: 7 inches
- B: 9 inches
- C: 3 inches
- D: 10 inches
Next, we will evaluate all possible combinations of three straws from the four available straws. The following combinations are possible:
- 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 using the triangle inequality:
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Combination A, B, C (7, 9, 3):
- \(7 + 3 > 9 \quad \Rightarrow \quad 10 > 9 \) (True)
- \(7 + 9 > 3 \quad \Rightarrow \quad 16 > 3 \) (True)
- \(9 + 3 > 7 \quad \Rightarrow \quad 12 > 7 \) (True)
- This combination forms a triangle.
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Combination A, B, D (7, 9, 10):
- \(7 + 9 > 10 \quad \Rightarrow \quad 16 > 10 \) (True)
- \(7 + 10 > 9 \quad \Rightarrow \quad 17 > 9 \) (True)
- \(9 + 10 > 7 \quad \Rightarrow \quad 19 > 7 \) (True)
- This combination forms a triangle.
-
Combination A, C, D (7, 3, 10):
- \(7 + 3 > 10 \quad \Rightarrow \quad 10 > 10 \) (False)
- This combination does NOT form a triangle.
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Combination B, C, D (9, 3, 10):
- \(9 + 3 > 10 \quad \Rightarrow \quad 12 > 10 \) (True)
- \(9 + 10 > 3 \quad \Rightarrow \quad 19 > 3 \) (True)
- \(3 + 10 > 9 \quad \Rightarrow \quad 13 > 9 \) (True)
- This combination forms a triangle.
After checking each combination, we find that the following combinations can form triangles:
- 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 straws, which are:
- Straws A, B, C (7, 9, 3)
- Straws A, B, D (7, 9, 10)
- Straws B, C, D (9, 3, 10)
In conclusion, Joanna can form 3 different triangles.
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