To determine how many different triangles Joanna can make with her four straws of different lengths, we need to use a fundamental rule in triangle geometry: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side (the triangle inequality theorem).
Let's say the lengths of the straws are \( a < b < c < d \). Joanna can choose combinations of three straws from these four. The combinations of three straws can be listed as follows:
- \( (a, b, c) \)
- \( (a, b, d) \)
- \( (a, c, d) \)
- \( (b, c, d) \)
Now, we need to check each combination against the triangle inequality.
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Combination \( (a, b, c) \):
- Check if \( a + b > c \)
- Check if \( a + c > b \) (which is always true since \( c > b \))
- Check if \( b + c > a \) (which is always true since \( c > a \))
If \( a + b > c \) holds, this combination can form a triangle.
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Combination \( (a, b, d) \):
- Check if \( a + b > d \)
- Check if \( a + d > b \) (which is always true since \( d > b \))
- Check if \( b + d > a \) (which is always true since \( d > a \))
If \( a + b > d \) holds, this combination can form a triangle.
-
Combination \( (a, c, d) \):
- Check if \( a + c > d \)
- Check if \( a + d > c \) (which is always true since \( d > c \))
- Check if \( c + d > a \) (which is always true since \( d > a \))
If \( a + c > d \) holds, this combination can form a triangle.
-
Combination \( (b, c, d) \):
- Check if \( b + c > d \)
- Check if \( b + d > c \) (which is always true since \( d > c \))
- Check if \( c + d > b \) (which is always true since \( d > b \))
If \( b + c > d \) holds, this combination can form a triangle.
To identify how many different triangles Joanna can make, we would need to know the actual lengths of the straws to evaluate these conditions.
Assuming you provide the lengths of the straws, we can verify these inequalities to find which combinations can indeed form triangles. If not, you can follow this method with the lengths you have to arrive at the correct answer.
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