What type of figure must a quadrilateral be if its diagonals are perpendicular bisectors of each other and are congruent?

The answer is A Square

The diagonals of a square are both perpendicular bisectors and congruent. This is unique to a square. Other parallelograms, such as rectangles, have congruent diagonals, and the diagonals of a rhombus are perpendicular bisectors, but the square is the only shape that satisfies both.