Understanding the Rhombus and its Properties

A rhombus is a special type of quadrilateral where all four sides are equal in length. It is also a parallelogram, which means its opposite sides are parallel, and opposite angles are equal. Key properties of a rhombus are:

• All four sides are congruent.
• The diagonals bisect each other at right angles (90°).
• The diagonals bisect the angles at the vertices.
• The area of a rhombus is given by the formula: Area $= \frac{1}{2} \times d_1 \times d_2$, where $d_1$ and $d_2$ are the lengths of the two diagonals.

In this problem, we are given the area of the rhombus and the length of one diagonal. We need to find the length of each side of the rhombus.

Calculating the Length of the Second Diagonal of the Rhombus

We are given:

• Area of the rhombus $= 24 \, \text{m}^2$
• Length of one diagonal, let's say $d_1 = 8 \, \text{m}$

Using the area formula for a rhombus:

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

Substitute the given values:

$24 = \frac{1}{2} \times 8 \times d_2$

$24 = 4 \times d_2$

To find $d_2$, divide both sides by 4:

$d_2 = \frac{24}{4}$

$d_2 = 6 \, \text{m}$

So, the length of the second diagonal is 6 m.

Finding the Side Length using the Pythagorean Theorem

The diagonals of a rhombus bisect each other at right angles. This property is crucial because it divides the rhombus into four congruent right-angled triangles. The legs of each right-angled triangle are half the lengths of the diagonals, and the hypotenuse of each triangle is the side of the rhombus.

Let the side length of the rhombus be $a$.

The lengths of the semi-diagonals are:

• Semi-diagonal 1: $\frac{d_1}{2} = \frac{8}{2} = 4 \, \text{m}$
• Semi-diagonal 2: $\frac{d_2}{2} = \frac{6}{2} = 3 \, \text{m}$

In one of the right-angled triangles, the two legs are 4 m and 3 m, and the hypotenuse is the side $a$ of the rhombus. We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($c^2 = a^2 + b^2$).

Applying the Pythagorean theorem:

$a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$

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

$a^2 = 16 + 9$

$a^2 = 25$

To find $a$, take the square root of both sides:

$a = \sqrt{25}$

$a = 5 \, \text{m}$

Therefore, the length of each side of the rhombus is 5 m.

Summary of Steps to Find Rhombus Side

Here’s a quick summary of how we solved the problem:

1. Used the area formula (Area $= \frac{1}{2} d_1 d_2$) to find the length of the second diagonal ($d_2$).
2. Calculated half the lengths of both diagonals ($d_1/2$ and $d_2/2$).
3. Applied the Pythagorean theorem ($a^2 = (d_1/2)^2 + (d_2/2)^2$) using the semi-diagonals as the legs of a right triangle and the side of the rhombus as the hypotenuse.
4. Calculated the value of the side length $a$.

Final Answer for Rhombus Side Length

Based on our calculations, the length of each side of the rhombus is 5 m.

Revision Table: Rhombus Formulas

Additional Information: Properties of Quadrilaterals

The rhombus is part of the family of quadrilaterals. Understanding other quadrilateral properties can help in geometry problems.

• Parallelogram: Opposite sides are parallel and equal. Opposite angles are equal. Diagonals bisect each other.
• Rectangle: A parallelogram with four right angles. Diagonals are equal.
• Square: A rectangle with four equal sides. Also a rhombus with four right angles. Diagonals are equal and bisect each other at right angles.
• Trapezoid (or Trapezium): A quadrilateral with at least one pair of parallel sides.
• Kite: Two pairs of adjacent sides are equal. Diagonals are perpendicular. One diagonal is the perpendicular bisector of the other.

The rhombus shares properties with parallelograms (diagonals bisect each other) and kites (diagonals are perpendicular). Its unique property is having all sides equal, which, combined with the perpendicular diagonals, leads to the relationship with the Pythagorean theorem used in this problem.