Question

Which kind of quadrilateral is indicated by the following statement? The diagonals are congruent, perpendicular, and bisect each other.

Short answer

The quadrilateral is a square.

Step-by-step solution

Identify Properties of Diagonals

The problem states that a quadrilateral has diagonals that are congruent, perpendicular, and bisect each other. We need to identify which quadrilaterals have these diagonal properties.

Review Quadrilateral Types

Recall the properties of common quadrilaterals: squares, rectangles, rhombuses, and kites. - Squares: Diagonals are congruent, perpendicular, and bisect each other. - Rectangles: Diagonals are congruent and bisect each other, but not necessarily perpendicular. - Rhombuses: Diagonals are perpendicular and bisect each other, but not necessarily congruent. - Kites: Diagonals are perpendicular but do not necessarily bisect or congruent.

Match Properties with Quadrilateral

The only common quadrilateral where the diagonals are congruent, perpendicular, and bisect each other is a square.

Key concepts

Diagonals of Quadrilaterals

Diagonals are line segments drawn from one vertex of a quadrilateral to the opposite vertex. The properties of these diagonals can reveal a lot about the type of quadrilateral they belong to.

• Congruent Diagonals: When diagonals are congruent, it means they have the same length.
• Perpendicular Diagonals: Diagonals are perpendicular when they intersect at a 90-degree angle.
• Bisecting Diagonals: Diagonals that bisect each other split each diagonal into two equal parts at the point of intersection.

For example, squares and rectangles both have congruent diagonals, but only squares have diagonals that are also perpendicular and bisect each other. The combination of these three properties is unique to squares in the family of common quadrilaterals.

Types of Quadrilaterals

Quadrilaterals come in various forms, each with distinct properties that define their class. Here is a concise guide:

• Squares: All four sides and angles are equal. Diagonals are congruent, perpendicular, and bisect each other.
• Rectangles: Opposite sides are equal with all angles measuring 90 degrees. Diagonals are congruent and bisect each other but are not perpendicular.
• Rhombuses: All sides are equal, and opposite angles are equal. Diagonals are perpendicular and bisect each other but are not congruent.
• Kites: Two pairs of adjacent sides are equal with one pair of opposite angles equal. Diagonals are perpendicular, but they do not necessarily bisect each other.

Understanding these basic properties can help you quickly determine what type of quadrilateral you are dealing with based on its diagonal properties.

Geometric Properties of Squares

Squares are one of the most symmetric and regularly shaped quadrilaterals. They have several key geometric properties:

• Equal Sides: All four sides are of equal length.
• Right Angles: Each angle is 90 degrees, making it a rectangle as well.
• Congruent and Perpendicular Diagonals: The diagonals are equal in length, intersect at right angles, and bisect each other.
• Symmetrical: Its high level of symmetry means each diagonal is both an axis of symmetry and divides the square into two congruent isosceles right triangles.

These properties make squares both unique and easily identifiable among quadrilaterals, especially when analyzing side lengths and diagonal behavior.