Understanding the Rhombus Problem

The question asks us to find the length of one side of a rhombus. We are given two pieces of information: the area of the rhombus and the sum of the lengths of its diagonals.

A rhombus is a quadrilateral with four sides of equal length. Its diagonals bisect each other at right angles. This property is key to relating the diagonals to the side length using the Pythagorean theorem.

Key Formulas for Rhombus Calculations

To solve this problem, we need to use the following formulas:

• Area of a rhombus: Area $= \frac{1}{2} \times d_1 \times d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.
• Pythagorean theorem: In a right-angled triangle with legs $a$ and $b$ and hypotenuse $c$, $a^2 + b^2 = c^2$. In a rhombus, the half-diagonals are the legs of a right triangle, and the side of the rhombus is the hypotenuse. So, $s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$, where $s$ is the side length.

Setting Up the Equations

We are given:

• Area $= 24 \text{ cm}^2$
• Sum of diagonals, $d_1 + d_2 = 14 \text{ cm}$

Using the area formula, we can write:

$\frac{1}{2} d_1 d_2 = 24$

$d_1 d_2 = 48 \quad (*)$

We also have the sum of diagonals:

$d_1 + d_2 = 14 \quad (**)$

We now have a system of two equations with two variables, $d_1$ and $d_2$.

Solving for the Diagonals

From equation $(**)$, we can express $d_2$ in terms of $d_1$: $d_2 = 14 - d_1$.

Substitute this into equation $(*)$:

$d_1 (14 - d_1) = 48$

$14d_1 - d_1^2 = 48$

Rearrange the terms to form a quadratic equation:

$d_1^2 - 14d_1 + 48 = 0$

We can solve this quadratic equation by factoring:

We need two numbers that multiply to 48 and add up to -14. These numbers are -6 and -8.

$(d_1 - 6)(d_1 - 8) = 0$

This gives us two possible values for $d_1$: $d_1 = 6$ or $d_1 = 8$.

If $d_1 = 6 \text{ cm}$, then from $d_2 = 14 - d_1$, we get $d_2 = 14 - 6 = 8 \text{ cm}$.

If $d_1 = 8 \text{ cm}$, then from $d_2 = 14 - d_1$, we get $d_2 = 14 - 8 = 6 \text{ cm}$.

So, the lengths of the diagonals are 6 cm and 8 cm (in any order).

Calculating the Side Length using the Pythagorean Theorem

Now that we have the lengths of the diagonals, we can find the side length of the rhombus. The diagonals of a rhombus divide it into four congruent right-angled triangles. The legs of each right triangle are half the lengths of the diagonals ($\frac{d_1}{2}$ and $\frac{d_2}{2}$), and the hypotenuse is the side of the rhombus ($s$).

Using the Pythagorean theorem: $s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$

Let's use $d_1 = 6$ cm and $d_2 = 8$ cm.

$s^2 = \left(\frac{6}{2}\right)^2 + \left(\frac{8}{2}\right)^2$

$s^2 = (3)^2 + (4)^2$

$s^2 = 9 + 16$

$s^2 = 25$

To find the side length $s$, take the square root of both sides:

$s = \sqrt{25}$

$s = 5 \text{ cm}$

The length of one side of the rhombus is 5 cm.

Summary of Steps

• Use the area formula to relate the product of diagonals to the area.
• Use the given sum of diagonals.
• Solve the resulting system of equations to find the lengths of the diagonals.
• Apply the Pythagorean theorem to the right triangle formed by the half-diagonals and the side to find the side length.

Revision Table: Rhombus Properties & Formulas

Additional Information: Connecting Rhombus, Square, and Parallelogram

A rhombus is a special type of parallelogram where all sides are equal. A square is a special type of rhombus where all angles are right angles (which also means its diagonals are equal). Understanding the relationships between these shapes helps in solving geometry problems.

In a rhombus, the diagonals create symmetry and provide a powerful way to calculate area and side length, especially when using the right-angle property for the Pythagorean theorem.