The shape that has diagonals that bisect each other at 90 degrees is a square. A square is a four-sided polygon with all sides equal in length and all angles equal to 90 degrees. One of the defining characteristics of a square is that its diagonals are equal in length and bisect each other at 90 degrees.

To understand why the diagonals of a square bisect each other at right angles, let’s consider the geometric properties of a square. Each side of a square is perpendicular to the adjacent sides, which means that all angles within a square are right angles. This perpendicularity of the sides is what leads to the diagonals of a square intersecting at 90 degrees.

When we draw the diagonals of a square, they divide the square into four congruent right triangles. These right triangles have two sides of equal length, which are the sides of the square, and one hypotenuse, which is the diagonal of the square. In a right triangle, the hypotenuse is always the longest side, and the other two sides are perpendicular to each other.

Since the diagonals of a square are equal in length and intersect at the center of the square, they divide each other into two congruent segments. These segments form the sides of the right triangles mentioned earlier.

When two lines intersect, they create vertical angles, which are congruent. In the case of the diagonals of a square, the vertical angles formed at the intersection are right angles, each measuring 90 degrees.

In practical terms, you can visualize the diagonals of a square by drawing a square on a piece of paper and then drawing lines connecting the opposite corners of the square. By measuring the lengths of the diagonals and the angles at their intersection, you will find that they are equal in length and intersect at 90 degrees.

Understanding the properties of a square and its diagonals is not only important in geometry but also has practical applications. For example, if you are constructing a square-shaped room, knowing that the diagonals bisect each other at 90 degrees can help you ensure that the corners are properly aligned and the room is perfectly square.

The diagonals of a square are equal in length and bisect each other at 90 degrees. This property is a characteristic of squares and can be explained by the perpendicularity of the sides of a square and the congruent right triangles formed by the diagonals.