Exploring Rhombus and its Theorems – Properties, examples, and Application

A rhombus is a parallelogram in which all sides are equal in length.  It is also known as a diamond or a lozenge.

Let's explore Rhombus and its Theorems – Properties, examples, and Applications in detail.....

Here are the properties of a rhombus:

All sides are equal: The most obvious property of a rhombus is that all four sides are equal in length. This means that if you know the length of one side, you can find the length of all the other sides.

AB=BC=CD=AD

Opposite angles are equal: Another property of a rhombus is that opposite angles are equal in measure. This means that if you know the measure of one angle, you can find the measure of the opposite angle.

∡A ≌ ∡C and ∡B ≌ ∡D

Diagonals bisect each other: The diagonals of a rhombus bisect each other at right angles. This means that they divide each other into two equal parts.

AO ≌ OC and BO ≌OD

Diagonals are perpendicular: The diagonals of a rhombus are perpendicular to each other. This means that they form a right angle where they intersect.

AC ⊥ BD So all angles at the intersection of the two diagonals are 90°

Rhombus and its Theorems

Theorem 1 : The diagonals of a rhombus are perpendicular to each other

Given: A rhombus ABCD whose diagonals are AC and BD intersect at O.
Prove that: ∠BOC = ∠DOC = ∠AOD = ∠AOB = 90°

Hence, ∠AOB = ∠BOC = ∠COD = ∠DOA = 90°

The Converse of the above theorem is also true ⇒ If the diagonals of a parallelogram are perpendicular, then it is a rhombus.

Symmetry: A rhombus has two lines of symmetry. These are the lines that pass through the opposite angles.

AC and BD are the two lines of symmetry

Area: The area of a rhombus can be found by multiplying the length of one diagonal by the length of the other diagonal and dividing the result by 2. That is, A = (d1 × d2) / 2, where A is the area and d1 and d2 are the lengths of the diagonals.

Perimeter: The perimeter of a rhombus can be found by multiplying the length of one side by 4. ie; P = 4s, where P is the perimeter and s is the length of one side.

Overall, a rhombus is a special type of parallelogram with four equal sides and some unique properties that make it useful in mathematics and geometry.

Theorems of rhombus with examples and formulae
There are several theorems related to the properties of a rhombus. Here are a few of them along with examples and formulae:

The Diagonal Theorem:

The diagonals of a rhombus bisect each other at right angles.

AO ≌ OC and BO ≌OD and AC ⊥ BD So all angles at the intersection of the two diagonals are 90°

Example: ABCD is a rhombus with diagonals of length 10 cm and 12 cm.

Find the length of each side.

Solution:

AC= 10 cm and BD= 12 cm

Using the Diagonal Theorem, we know that the diagonals bisect each other at right angles.

Therefore, each diagonal is split into two parts AO = OC = 5cm and

BO =DO = 6 cm

To  find the length of the side of the rhombus:

We can then use the Pythagorean theorem in triangle AOB suppose

a² + b² = c²

where a = 5 cm, b = 6 cm, and c is the length of one side. Solving for c, we get:

c = √(5² + 6²) = √61

So the length of each side of the rhombus is √61 cm.

The Side and Angle Theorems:
In a rhombus, opposite angles are equal, and adjacent angles are supplementary.

Let's say we have a rhombus ABCD with a side length of 6 cm. Using the Side and Angle Theorems, we know that opposite angles are equal. We also know that adjacent angles are supplementary, so each pair of adjacent angles adds up to 180 degrees.

AB= BC=CD=AD = 6 cm

∡A =∡C and ∡B= ∡D and

∡A + ∡D = 180°

∡A + ∡B = 180°

∡B + ∡C=180°

∡C + ∡D=180°

The Area Formula: The area of a rhombus can be found by multiplying the lengths of the diagonals and dividing by 2.

Example: PQRS is a rhombus with diagonals of length 8 cm and 10 cm. Find the area of the rhombus.

Solution :

Diagonals PR =8cm and QS = 10 cm

Using the area formula, we can find the area of the rhombus:

A = (d1 x d2) / 2

where d1 and d2 are the lengths of the diagonals. Substituting the values we have, we get:

A = (8 x 10) / 2 = 40

So the area of the rhombus is 40 square units.

The Perimeter Formula: The perimeter of a rhombus can be found by multiplying the length of one side by 4.

Example: Let's say we have a rhombus with a side length of 7 cm. Using the perimeter formula, we can find the perimeter of the rhombus:

P = 4s

where s is the length of one side. Substituting in the value we have, we get:

P = 4 x 7 = 28

So the perimeter of the rhombus is 28 cm.

Example :
ABCD is a rhombus with ∠ABC = 56. Determine ∠ACD.
Solution : ABCD is a rhombus.
∠ ABC = ∠ADC ( Opposite angles are equal)
∠ADC = 560
∴ ∠ ODC = ½ ∠ADC ( Diagonals of rhombus bisects the angle)
⇒ ∠ODC = ½ x 56
⇒ ∠ODC = 28°
∠OCD + ∠ODC + ∠COD = 180 ( In ΔOCD, the sum of all the angles in a triangle is 180)
∠OCD + 28 + 90 = 180
⇒ ∠OCD + 118 = 180
⇒ ∠OCD = 180 -118
∠OCD = 62°

These are just a few of the theorems related to the properties of a rhombus. There are many more that can be used to solve problems involving this special type of parallelogram.

Rhombuses have a variety of practical and mathematical applications; here are some of the most common uses of a rhombus:

Jewellery: Rhombus-shaped gems or diamond cuts are a popular choice for engagement rings, earrings, and other types of jewelry. The symmetry and elegance of the rhombus shape make it an attractive option for many people.

Tiling: Rhombuses can be used to create beautiful and unique tiled patterns for floors and walls. The symmetrical shape allows for interesting and intricate designs that can be easily repeated across a surface.

Mathematics: Rhombuses are a fundamental shape in geometry and can be used to teach concepts like symmetry, angles, and area. They are also used in more advanced mathematical concepts like the Pythagorean theorem and trigonometry.

Engineering: Rhombuses are often used in the design of trusses, bridges, and other structures. The equal length of all four sides and the perpendicular diagonals make the rhombus a stable and reliable shape for building.

Art: Rhombuses can be used in art to create geometric shapes and patterns. Many artists use rhombuses as a starting point for their designs and incorporate them into their work in various ways.

Navigation: Rhombuses can be used in navigation to calculate distances and bearings. The shape of a rhombus can help sailors and pilots calculate the shortest distance between two points on a map.

Overall, the rhombus is a versatile shape that has many practical and mathematical applications. Its unique properties make it a useful tool in a variety of fields, from art and design to engineering and mathematics.

People also want to know…..

What are the opposite sides of a rhombus?

In a rhombus, opposite sides are parallel and congruent. This means that the opposite sides of a rhombus have the same length and are parallel to each other. Therefore, any pair of opposite sides of a rhombus can be referred to as opposite sides.

What are the formulae for rhombuses?

Here are the formulae for some of the properties of a rhombus:

The perimeter of a Rhombus:

The perimeter of a rhombus can be found by multiplying the length of one side by 4.

P = 4s

where P is the perimeter of the rhombus and s is the length of one side.

Area of a Rhombus:

The area of a rhombus can be found by multiplying the lengths of the diagonals and dividing by 2.

A = (d1 x d2) / 2

where A is the area of the rhombus, d1, and d2 are the lengths of the diagonals.

Length of a Side of a Rhombus:

The length of a side of a rhombus can be found using the Pythagorean theorem.

a²+ b² = c²

where a and b are half of the lengths of the diagonals and c is the length of a side of the rhombus.

c = √(a² + b²)

The diagonal of a Rhombus:

The length of the diagonals of a rhombus can be found using the formula:

d = √(s² + t²)

where d is the length of the diagonal, and s and t are half of the lengths of the sides of the rhombus.

These formulae can be used to solve various problems related to rhombuses in geometry and mathematics.

What are the angles in a rhombus?

In a rhombus, opposite angles are congruent and adjacent angles are supplementary. This means that:

Opposite angles of a rhombus are equal in measure.

The adjacent angles of a rhombus are supplementary, meaning they add up to 180 degrees.

Therefore, if one angle of a rhombus is known, then all the other angles can be calculated.

Example: In rhombus PQRS, ∡ P= 60° then find the measure of remaining angles of a rhombus.

Solution : ∡P= 60°(given)

∡R= 60° ( opposite angles are congruent)

∡P + ∡S= 180 (adjacent angles are supplementary)

60 + ∡S= 180

∡S= 180-60 =120

∡S= 120°

m∡S = m∡Q (opposite angles)

m∡Q = 120°

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