A Comprehensive Guide to the Properties of Rhombuses

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:

1. A rhombus has diagonals measuring \(12\) units and \(16\) units. Compute its area.
2. If one side of a rhombus is \(9\) units long, determine its perimeter.
3. 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:

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