Properties of Rectangles and Rhombuses

Introduction to quadrilaterals

Before learning about the properties of rectangles and rhombuses, let us first understand what quadrilaterals are in geometry. Quadrilaterals are polygons with four sides, four vertices, and four angles. There are many types of quadrilaterals, including squares, trapezoids, rectangles, and rhombuses, each of which has its own distinct properties.

What is a rectangle?

A rectangle is a type of quadrilateral where each angle is a right angle, meaning they are 90 degrees. The opposite sides of a rectangle are equal in length and parallel. Rectangles are very common in daily life and are used in things like windows, books, and screens.

Suppose you have a rectangle ABCD with sides AB, BC, CD and DA. The properties of a rectangle can be expressed in the following ways:
AB = CD
BC = DA
∠A = ∠B = ∠C = ∠D = 90°

Visual example of a rectangle

Diagonal properties of rectangles

A notable property of rectangles is that the diagonals are equal in length. The diagonals of a rectangle bisect each other, creating two right-angled triangular segments within the rectangle.

If you have a rectangle ABCD, then the diagonals AC and BD will satisfy the following:
AC = BD

For any rectangle:
AC = √(AB² + BC²) 
Since AB = A and BC = B in an axis-aligned rectangle, 
ac = √(a² + b²) 

Real life examples of rectangles

Rectangles are everywhere! Think of a book cover or a computer screen. These objects often have the properties of rectangles.

What is a rhombus?

A rhombus is another type of quadrilateral with all four sides of equal length. Unlike rectangles, the angles cannot be 90 degrees. However, the opposite angles in a rhombus are equal, and its diagonals intersect at right angles.

Consider a rhombus PQRS whose sides are PQ, QR, RS and SP:
PQ = QR = RS = SP
∠P = ∠R
∠Q = ∠S

Visual example of a rhombus

Diagonal properties of rhombus

Like rectangles, rhombuses also have interesting diagonal properties. The diagonals of a rhombus bisect each other at right angles, and they are not necessarily equal.

If we consider the diagonals PR and QS of rhombus PQRS:
Diagonals PR and QS intersect at 90°

For any rhombus:
Pr ⊥ Qs
Each diagonal divides the rhombus into two isosceles triangles

Real life examples of rhombus

A common example of a rhombus is the diamond shape, which is often found in design patterns and jewellery.

Similarities between rectangles and rhombuses

Both rectangles and rhombuses have some similarities. Both have opposite sides of equal length. They are also called parallelograms, which means that both have opposite sides that are parallel.

Difference between rectangles and rhombuses

While they share similarities, rectangles and rhombuses are different in several ways. Rectangles always have four right angles, while rhombuses may not. All four sides of a rhombus are equal, while rectangles only have opposite sides equal.

Summary

Understanding the properties of rectangles and rhombuses helps to identify them and differentiate between the different types of quadrilaterals. Rectangles are characterized by having equal diagonals and right angles, while rhombuses are characterized by having equal side lengths and diagonals that intersect at right angles. These properties make them a unique and playable part of the great quadrilateral family.

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