To determine the greatest distance between a vertex of a square inscribed in another square, one must consider the dimensions of both squares. The inner square has a perimeter of 20, giving it a side length of 5, while the outer square, with a perimeter of 28, has a side length of 7. The maximum distance between a vertex of the inner square and a vertex of the outer square occurs when the inner square is positioned diagonally within the outer square. The discussion clarifies that an inscribed square typically touches the sides of the outer square at its vertices, rather than along the base. Understanding these geometric relationships is crucial for solving the problem effectively.
