How do you prove that the diagonals of a rhombus are perpendicular?

To demonstrate that the diagonals of a rhombus are perpendicular, we can use some properties of triangles and the characteristics of a rhombus.

Key Properties of a Rhombus

A rhombus is a type of quadrilateral where all sides are equal in length. Additionally, the diagonals of a rhombus have some unique properties:

• They bisect each other.
• They bisect the angles of the rhombus.

Step-by-Step Proof

Let’s label the rhombus as ABCD, with diagonals AC and BD intersecting at point O.

1. Since the diagonals bisect each other, we have AO = OC and BO = OD.
2. Triangles AOB and COD are formed by the diagonals.
3. In triangles AOB and COD:
• AB = CD (sides of the rhombus)
• AO = OC (diagonal bisected)
• BO = OD (diagonal bisected)
4. By the Side-Side-Side (SSS) congruence criterion, triangles AOB and COD are congruent.

Conclusion of the Proof

Since triangles AOB and COD are congruent, the angles AOB and COD are equal. Because these angles are adjacent and form a linear pair, they must add up to 180 degrees. Therefore, each angle must be 90 degrees, proving that the diagonals AC and BD are perpendicular to each other.

This property is essential in understanding the unique characteristics of a rhombus and can be applied in various geometric problems.