To determine how many different triangles Joanna can create with the straws of lengths 7 inches, 9 inches, 3 inches, and 10 inches, we first need to apply the triangle inequality theorem. According to this theorem, for any three lengths \( a \), \( b \), and \( c \) to form a triangle, the following conditions must be satisfied:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
We will check all combinations of three straws from the four available straws (A, B, C, and D based on their lengths) and see if they meet these conditions.
Combinations to Consider:
- Straws A, B, C: Lengths 7, 9, 3
- Straws A, B, D: Lengths 7, 9, 10
- Straws A, C, D: Lengths 7, 3, 10
- Straws B, C, D: Lengths 9, 3, 10
We will check each combination to see if they can form a triangle.
Checking Combinations:
-
Combination (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)
- Valid Triangle
-
Combination (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)
- Valid Triangle
-
Combination (A, C, D): 7, 3, 10
- Check \( 7 + 3 > 10 \) → \( 10 > 10 \) (False)
- Not a valid triangle
-
Combination (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)
- Valid Triangle
Summary:
The valid combinations that form triangles are:
- Straws A, B, C (Lengths 7, 9, 3)
- Straws A, B, D (Lengths 7, 9, 10)
- Straws B, C, D (Lengths 9, 3, 10)
Total Number of Valid Triangles:
Joanna can make 3 different triangles.
- Triangle 1: Using 7, 9, 3
- Triangle 2: Using 7, 9, 10
- Triangle 3: Using 9, 3, 10
Thus, the final answer is that Joanna can make 3 different triangles and those combinations of straws used are (7, 9, 3), (7, 9, 10), and (9, 3, 10).
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