Joanna has four straws of different lengths. The table shows the lengths of the straws.

Joanna makes as many triangles as she can using combinations of three of these straws. Determine how many different triangles she can make. What combinations of straws does she use?

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(4 points)

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1 answer

To determine the number of triangles that Joanna can form using her four straws of different lengths, we should first consider the given lengths. While the specific lengths are not mentioned in the problem, we can denote them as \( a \), \( b \), \( c \), and \( d \), with the understanding that \( a < b < c < d \).

For any three lengths to form a triangle, they must satisfy the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we will systematically assess all combinations of three straws selected from the four available.

The possible combinations of three lengths from the four straws are: \( (a, b, c) \), \( (a, b, d) \), \( (a, c, d) \), and \( (b, c, d) \). We need to check each combination against the triangle inequality conditions.

For the combination \( (a, b, c) \): The inequalities \( a + b > c \), \( a + c > b \), and \( b + c > a \) must hold true. Given the order of lengths, it is likely that this combination will form a triangle.
For the combination \( (a, b, d) \): We must check if \( a + b > d \). Here, since \( d \) is the longest straw, this inequality is likely to fail, and thus they cannot form a triangle.
For the combination \( (a, c, d) \): Checking \( a + c > d \) is necessary. Since \( d \) is the longest straw, this combination will also likely not satisfy the triangle condition.
Lastly, for \( (b, c, d) \): We need to check if \( b + c > d \). Depending on the specific values of the lengths, this inequality could go either way, but if \( b \) and \( c \) are sufficiently larger than \( d \), a triangle could be formed.

After reviewing all combinations based on the triangle inequality, we can conclude the combinations that do form triangles. Therefore, assuming favorable lengths for certain combinations, Joanna may be able to form either one or two valid triangles depending on specific lengths provided. It is crucial to check the triangle inequalities for exact lengths to confirm the number of triangles successfully made.