The diagonals of a square are perpendicular bisectors because of the properties of the square's symmetry and geometry. Here are the reasons why:
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Equal Length: In a square, the diagonals are of equal length because all sides are equal and the angles between sides are right angles. The diagonals connect opposite corners (vertices) of the square, and due to the Pythagorean theorem, both diagonals have the same length.
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Bisection: The diagonals intersect at the center of the square, dividing each diagonal into two equal segments. This point of intersection is equidistant from the four corners of the square, confirming that each diagonal bisects the other.
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Perpendicularity: Since the angles in a square are all 90 degrees, the diagonals intersect each other at an angle of 90 degrees. Thus, the diagonals are perpendicular to each other.
To summarize, the properties that lead to the conclusion that the diagonals of a square are perpendicular bisectors stem from the square's symmetrical structure, equal side lengths, and right angles at each vertex.
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