When proving a square, how many distances must be calculated?

To prove that a quadrilateral is a square, it is necessary to demonstrate that it possesses specific properties: all sides are equal in length, and all angles are right angles.

Calculating four distances is crucial because a square is characterized by its combination of equal sides and right angles. By calculating the distances between four vertices, we can establish whether all four sides are of equal length. Additionally, checking the diagonals—two extra distances—can help confirm that the diagonals are also equal, which is a property unique to squares among quadrilaterals.

In a square, the lengths of the sides are equal, and the diagonals are equal and bisect each other at right angles. Therefore, while it's possible to calculate only the distances of the sides initially to confirm their equality, ensuring that both the side lengths and the diagonal lengths meet the necessary criteria solidifies the proof that the figure is indeed a square. Thus, calculating four distances allows a comprehensive verification of all necessary properties to confirm that the quadrilateral is a square.