To determine how many different triangles Joanna can make with the straws of lengths 3 inches (C), 7 inches (A), 9 inches (B), and 10 inches (D), we will apply the triangle inequality theorem. The triangle inequality states that for any three lengths \(a\), \(b\), and \(c\), a triangle can be formed if the following conditions are satisfied:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
We'll evaluate all possible combinations of three straws from the list:
-
Combination 1: (3, 7, 9)
- \(3 + 7 = 10\) (not greater than 9)
- Cannot form a triangle
-
Combination 2: (3, 7, 10)
- \(3 + 7 = 10\) (not greater than 10)
- Cannot form a triangle
-
Combination 3: (3, 9, 10)
- \(3 + 9 = 12 > 10\)
- \(3 + 10 = 13 > 9\)
- \(9 + 10 = 19 > 3\)
- Can form a triangle
-
Combination 4: (7, 9, 10)
- \(7 + 9 = 16 > 10\)
- \(7 + 10 = 17 > 9\)
- \(9 + 10 = 19 > 7\)
- Can form a triangle
After analyzing all combinations of three lengths, the only combinations that can form triangles are:
- (3, 9, 10)
- (7, 9, 10)
Therefore, Joanna can make a total of 2 different triangles using the straws of lengths 3, 9, and 10 inches, and 7, 9, and 10 inches.
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