# A Comprehensive Guide to the Properties of Rhombuses

A rhombus, often viewed as a "slanted square," is a captivating quadrilateral that brings with it a set of distinctive properties. Every side of a rhombus is of equal length, making it a special type of parallelogram. While it shares some features with other parallelograms, its equilateral nature and intriguing internal angles make it a standout figure in geometry. As we delve deeper, we'll discover the rich geometric attributes and characteristics that make rhombuses both fascinating and fundamental in the realm of shapes.

## Step-by-step Guide: Properties of The Rhombus

**Definition of a Rhombus**:

A rhombus is a four-sided polygon (quadrilateral) where all four sides are of equal length. This ensures that opposite angles of a rhombus are equal.

**Parallel Opposite Sides**:

The opposite sides of a rhombus are always parallel to each other, making it a type of parallelogram.

**Equal Diagonals**:

While the diagonals of a rhombus are not necessarily equal in length, they do bisect each other at right angles. This means that they divide each other into two equal parts.

**Area**:

The area of a rhombus can be determined using its diagonals. If \(d_1\) and \(d_2\) represent the lengths of the diagonals, then the area \(A\) is given by:

\( A = \frac{d_1 \times d_2}{2}\)

**Perimeter**:

As all sides of a rhombus are equal, if a side is of length \(s\), the perimeter \(P\) is:

\( P = 4s \)

**Acute and Obtuse Angles**:

A rhombus can have both acute and obtuse angles. The opposite angles are always equal. If one angle is acute, the angle adjacent to it will be obtuse, and vice versa.

**Examples**

**Example 1:**

Finding the Area of a Rhombus.

Given: A rhombus with diagonals measuring \(10\) units (longer diagonal) and \(8\) units (shorter diagonal).

**Solution**:

The area of a rhombus can be determined using the formula:

\(A = \frac{d_1 \times d_2}{2}\)

Where \(d_1\) and \(d_2\) are the lengths of the diagonals. Essentially, the area is half the product of its diagonals. This formula stems from the fact that the diagonals of a rhombus are perpendicular bisectors of each other, dividing the rhombus into four congruent right triangles.

Area: Using the formula, we get:

\( A = \frac{d_1 \times d_2}{2} = \frac{10 \times 8}{2} = 40 \) square

**Example 2:**

Calculating the Perimeter of a Rhombus.

Given: A rhombus with a side length of \(7\) units.

**Solution**:

The perimeter of a rhombus, or any polygon for that matter, is simply the sum of the lengths of its sides. Since all sides of a rhombus are of equal length, the perimeter can be calculated by multiplying one side by four.

Perimeter: Using the concept above, we find:

\( P = 4 \times \text{side length} = 4 \times 7 = 28 \) units

### Practice Questions:

- A rhombus has diagonals measuring \(12\) units and \(16\) units. Compute its area.
- If one side of a rhombus is \(9\) units long, determine its perimeter.
- One diagonal of a rhombus is half the length of the other. If the longer diagonal is \(20\) units, find the area of the rhombus.

**Answers:**

**Area**: \( A = \frac{12 \times 16}{2} = 96 \) square units.**Perimeter**: \( P = 4 \times 9 = 36 \) units.- The shorter diagonal is \(10\) units.
**Area**: \( A = \frac{20 \times 10}{2} = 100 \) square units.

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