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Answer:

To determine if a quadrilateral with the given side lengths is possible, we can apply the triangle inequality theorem. According to this theorem, in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Let's analyze the possible combinations of three sides to see if they satisfy the triangle inequality:

1. 2cm, 3cm, and 5cm:

The sum of the two smaller sides (2cm + 3cm = 5cm) is equal to the length of the largest side (5cm). This combination is valid.

2. 2cm, 3cm, and 8cm:

The sum of the two smaller sides (2cm + 3cm = 5cm) is less than the length of the largest side (8cm). This combination is not valid.

3. 2cm, 5cm, and 8cm:

The sum of the two smaller sides (2cm + 5cm = 7cm) is equal to the length of the largest side (7cm). This combination is valid.

4. 3cm, 5cm, and 8cm:

The sum of the two smaller sides (3cm + 5cm = 8cm) is equal to the length of the largest side (8cm). This combination is valid.

Since there are valid combinations of three sides that satisfy the triangle inequality, it is possible to form a quadrilateral with side lengths of 2cm, 3cm, 5cm, and 8cm.

To determine the number of distinct quadrilaterals, we need to consider the possible positions of the diagonal. Since a diagonal divides a quadrilateral into two triangles, we have two cases to consider:

Case 1: The diagonal is between the sides of length 2cm and 8cm.

In this case, the diagonal has a length of 7cm, which satisfies the triangle inequality for both triangles. So, there is only one distinct quadrilateral in this case.

Case 2: The diagonal is not between the sides of length 2cm and 8cm.

In this case, the diagonal cannot have a length of 7cm because the sum of the lengths of the two sides adjacent to the diagonal is less than 7cm. Therefore, there are no distinct quadrilaterals in this case.

In conclusion, there is only one distinct quadrilateral possible with side lengths of 2cm, 3cm, 5cm, and 8cm, and the diagonal has a length of 7cm.