Finding the Area of a Rhombus Using Diagonal Ratios

In geometry, a rhombus is a fascinating quadrilateral with all sides of equal length. One of the key properties of a rhombus is that its diagonals bisect each other at right angles. This property proves particularly useful when determining the area of a rhombus, especially when the lengths of the diagonals are provided in a given ratio. In this article, we will walk through a detailed step-by-step process to compute the area of a rhombus given specific diagonal ratios and side lengths.

Understanding the Properties of a Rhombus

A rhombus is characterized by:

All four sides are equal in length. The diagonals intersect at right angles (90 degrees). Each diagonal bisects the other.

These properties allow us to utilize the Pythagorean theorem to establish relationships between the side length and the diagonals of the rhombus.

Step-by-Step Calculation

Let's consider a rhombus with a side length of 15 cm, where the diagonals are in the ratio 3:4. We will use this information to find the area of the rhombus.

Step 1: Understand the Properties

Side length of the rhombus, s 15 cm. The diagonals are in the ratio 3:4.

Step 2: Express the Diagonals Using the Given Ratio

Let the lengths of the diagonals be d_1 and d_2.

Given that d_1 : d_2 3 : 4, we can express the diagonals as:

d_1 3x d_2 4x

Step 3: Use the Pythagorean Theorem

The relationship between the side length, s, and the diagonals, d_1 and d_2, can be derived from the Pythagorean theorem:

s sqrt{left(frac{d_1}{2}right)^2 left(frac{d_2}{2}right)^2}

Substituting the expressions for d_1 and d_2, we get:

15 sqrt{left(frac{3x}{2}right)^2 left(frac{4x}{2}right)^2}

This simplifies to:

15 sqrt{left(frac{3x}{2}right)^2 4x^2}

15 sqrt{frac{9x^2}{4} 4x^2}

15 sqrt{frac{9x^2 16x^2}{4}}

15 sqrt{frac{25x^2}{4}}

15 frac{5x}{2}

Step 4: Solve for x

Multiplying both sides by 2:

30 5x

x 6

Step 5: Calculate the Lengths of the Diagonals

Substitute x back into the expressions for the diagonals:

d_1 3x 36 18 text{ cm} d_2 4x 46 24 text{ cm}

Step 6: Calculate the Area of the Rhombus

The area, A, of a rhombus can be calculated using the formula:

A frac{1}{2} times d_1 times d_2

Substituting the values of the diagonals:

A frac{1}{2} times 18 times 24

A frac{1}{2} times 432 216 text{ cm}^2

Conclusion

The area of the rhombus is 216 square centimeters.

Alternative Solutions

For a rhombus with a side length of 20 cm and diagonals in the ratio 3:4:

Let one diagonal be 3x and the other 4x. Using the Pythagorean theorem:

20^2 left(frac{3x}{2}right)^2 left(frac{4x}{2}right)^2

400 frac{9x^2}{4} 4x^2

400 frac{9x^2 16x^2}{4}

400 frac{25x^2}{4}

x^2 frac{400}{6.25}

x^2 64

x 8

Diagonals are 24 and 32.

Area of rhombus frac{24 times 32}{2} 384 text{ cm}^2

For a rhombus with a side length of 15 cm and diagonals in the ratio 3:4:

Let the diagonals be 3x and 4x. Solving the equation:

3x/2^2 4x^2 400

9x^2/4 16x^2 400

25x^2/4 400

5x/2 20

x 8