Final answer:
According to the triangle inequality theorem, the length of any side of a triangle must be less than the sum of the lengths of the other two sides. Therefore, in the context of this question, all the options provided - x cm, 2x cm, 3x/2 cm, and 5x/2 cm - can represent the length of the third side of the triangle as all of them are less than 6x cm.
Step-by-step explanation:
The subject of this problem is mathematics, specifically geometry and the properties of triangles. According to the triangle inequality theorem, the length of any side of a triangle must be less than the sum of the lengths of the other two sides.
Given that two sides of the triangle measure 2x cm and 4x cm, the sum of these two sides is 6x cm. Therefore, the length of the third side must be less than 6x cm.
Looking at the options: A. x cm, B. 2x cm, C. 3x/2 cm, and D. 5x/2 cm, we can see that all the lengths are smaller than 6x cm. Thus, all of them can represent the lengths of the third side given that x is a positive integer.
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