Final answer:
Given that the diagonals of quadrilateral TRUE are perpendicular bisectors of each other, the quadrilateral must be a square because it satisfies the properties of both a rhombus (equal sides) and a rectangle (right angles), thus making it option 3) Square.
Step-by-step explanation:
In the context of a quadrilateral TRUE, where diagonals TU and RE are stated to be perpendicular bisectors of each other, let's determine the nature of the quadrilateral. Perpendicular bisectors are lines or segments that intersect at right angles (90°) and bisect each other at their midpoint. This scenario means that each diagonal divides the other into two equal segments, and they intersect at a right angle.
For a quadrilateral with such properties, there are certain traits that it would display:
- It must have all four sides of equal length, given that the diagonals bisect each other; this is inherently a characteristic of a rhombus.
- It must also have four right angles, because if the diagonals are perpendicular, they form 90° angles where they intersect, which is a defining feature of a rectangle.
Since the quadrilateral satisfies both the properties of a rhombus and a rectangle, the only quadrilateral that is both a rhombus and a rectangle is a square. Therefore, the quadrilateral TRUE, with perpendicular bisecting diagonals, must be a square.
The mention correct option in final answer is therefore option 3) Square. The quadrilateral TRUE cannot simply be a rectangle or just a rhombus because these definitions alone do not satisfy both conditions simultaneously. In a square, however, these conditions are naturally met. Thus, you should choose only one option, and in this case, the correct choice is a square.