Understanding and Solving Problems related to Rhombuses: Properties, Area, Perimeter – A Comprehensive Guide

Rhombus

A rhombus is a polygon with four sides of equal length

A rhombus is a polygon with four sides of equal length. It is a special type of parallelogram where all sides are congruent. In addition, opposite angles are equal, and the diagonals are perpendicular bisectors of each other.

To understand and solve problems related to rhombuses, let’s look at some key properties:

1. Side Length: Since all sides of a rhombus have the same length, let’s denote it as “s”.

2. Diagonals: Diagonals of a rhombus are important to consider. They bisect each other at a 90-degree angle, forming four congruent right triangles. The length of each diagonal can be represented as “d_1” and “d_2”.

3. Area: The area of a rhombus can be calculated by multiplying the lengths of its diagonals and dividing by 2. So, the formula for the area “A” is A = (d_1 * d_2) / 2.

4. Perimeter: The perimeter of a rhombus can be found by adding up all four sides, as the sides are equal. Therefore, the perimeter “P” is P = 4s.

5. Interior Angles: Since a rhombus is a parallelogram, opposite angles are congruent. All angles in a rhombus are equal to each other, but we need additional information to determine the exact angle size.

Now, let’s solve an example problem using this information:

Problem:
The diagonals of a rhombus measure 10 cm and 12 cm. Find its area and perimeter.

Solution:
Given: d_1 = 10 cm, d_2 = 12 cm.

1. To calculate the area (A), we use the formula A = (d_1 * d_2) / 2.
Substituting the given values, A = (10 cm * 12 cm) / 2 = 60 cm^2.
Therefore, the area of the rhombus is 60 square centimeters.

2. To find the perimeter (P), we use the formula P = 4s.
Since all sides are equal, we just need to find the length of one side.
Using the Pythagorean theorem within one of the congruent right triangles formed by the diagonals, we have:
(1/2 * d_1)^2 + (1/2 * d_2)^2 = s^2
(1/2 * 10 cm)^2 + (1/2 * 12 cm)^2 = s^2
5 cm^2 + 6 cm^2 = s^2
11 cm^2 = s^2
Taking the square root of both sides, we find s ≈ 3.32 cm.
Now, substituting this s into the perimeter formula, we get P = 4 * 3.32 cm = 13.28 cm.
Therefore, the perimeter of the rhombus is approximately 13.28 centimeters.

In conclusion, we have found that the area of the given rhombus is 60 square centimeters, and its perimeter is approximately 13.28 centimeters.

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