Properties of Rhombus
A rhombus is a quadrilateral in which all four sides are equal in length. It is a special type of parallelogram, so it inherits all the properties of a parallelogram — opposite sides are parallel, opposite angles are equal, and diagonals bisect each other. However, a rhombus has additional properties that make it unique.
The word "rhombus" comes from the Greek word rhombos, meaning a spinning top or a shape that can be spun. In everyday life, the diamond shape on playing cards is a rhombus. Kite shapes, certain floor tiles, and traffic signs also use the rhombus shape.
In Class 8, you will study the specific properties of a rhombus — particularly how its diagonals behave. Understanding rhombus properties is essential for solving problems in the chapter "Understanding Quadrilaterals" and also for mensuration problems involving area of a rhombus.
What is Properties of Rhombus?
Definition: A rhombus is a parallelogram in which all four sides are equal.
A rhombus ABCD has the following defining features:
- AB = BC = CD = DA (all sides are equal)
- AB || CD and BC || DA (opposite sides are parallel)
- It is a special case of a parallelogram
Important: Every rhombus is a parallelogram, but every parallelogram is NOT a rhombus. A parallelogram becomes a rhombus only when all four sides are equal.
Properties of Rhombus Formula
Key Formulas for a Rhombus:
Area = (1/2) x d₁ x d₂
Where:
- d₁ = length of first diagonal
- d₂ = length of second diagonal
Also:
Perimeter = 4 x side
Relationship between side and diagonals:
side² = (d₁/2)² + (d₂/2)²
This follows from the Pythagoras theorem, since the diagonals bisect each other at right angles.
Derivation and Proof
Why do the diagonals of a rhombus bisect each other at right angles?
Step 1: A rhombus is a parallelogram. In any parallelogram, the diagonals bisect each other. So the diagonals of a rhombus bisect each other.
Step 2: Let the diagonals AC and BD of rhombus ABCD intersect at point O.
Since the diagonals bisect each other: OA = OC and OB = OD.
Step 3: Consider triangles AOB and COB:
- OA = OC (diagonals bisect each other)
- OB = OB (common side)
- AB = CB (all sides of rhombus are equal)
By SSS congruence, triangle AOB is congruent to triangle COB.
Step 4: Therefore, angle AOB = angle COB (corresponding parts of congruent triangles).
But angle AOB + angle COB = 180 degrees (linear pair).
So angle AOB = angle COB = 90 degrees.
Conclusion: The diagonals of a rhombus bisect each other at right angles (90 degrees).
Types and Properties
The properties of a rhombus can be grouped into the following categories:
1. Side Properties:
- All four sides are equal in length.
- Opposite sides are parallel.
2. Angle Properties:
- Opposite angles are equal.
- Adjacent angles are supplementary (add up to 180 degrees).
- The diagonals bisect the vertex angles (each diagonal divides the angle at the vertex into two equal parts).
- Diagonals bisect each other (cut each other into two equal halves).
- Diagonals bisect each other at right angles (90 degrees).
- Diagonals are generally NOT equal in length (unless the rhombus is a square).
- Each diagonal divides the rhombus into two congruent triangles.
- The two diagonals together divide the rhombus into four congruent right triangles.
4. Symmetry Properties:
- A rhombus has 2 lines of symmetry (along each diagonal).
- A rhombus has rotational symmetry of order 2 (it looks the same after a 180-degree rotation).
5. Special Cases:
- A square is a rhombus in which all angles are 90 degrees.
- Every square is a rhombus, but not every rhombus is a square.
Solved Examples
Example 1: Example 1: Finding the perimeter of a rhombus
Problem: One side of a rhombus is 13 cm. Find its perimeter.
Solution:
Given:
- Side of rhombus = 13 cm
Using the formula:
- Perimeter = 4 x side
- Perimeter = 4 x 13
- Perimeter = 52 cm
Answer: The perimeter of the rhombus is 52 cm.
Example 2: Example 2: Finding the area using diagonals
Problem: The diagonals of a rhombus are 16 cm and 12 cm. Find its area.
Solution:
Given:
- d₁ = 16 cm
- d₂ = 12 cm
Using the formula:
- Area = (1/2) x d₁ x d₂
- Area = (1/2) x 16 x 12
- Area = (1/2) x 192
- Area = 96 cm²
Answer: The area of the rhombus is 96 cm².
Example 3: Example 3: Finding the side using diagonals
Problem: The diagonals of a rhombus are 10 cm and 24 cm. Find the length of each side.
Solution:
Given:
- d₁ = 10 cm, so d₁/2 = 5 cm
- d₂ = 24 cm, so d₂/2 = 12 cm
The diagonals of a rhombus bisect each other at right angles. This forms a right triangle with legs 5 cm and 12 cm.
Using Pythagoras Theorem:
- side² = 5² + 12²
- side² = 25 + 144
- side² = 169
- side = 13 cm
Answer: Each side of the rhombus is 13 cm.
Example 4: Example 4: Finding an angle
Problem: In a rhombus PQRS, angle P = 120 degrees. Find all other angles.
Solution:
Given:
- Angle P = 120 degrees
Using properties of a rhombus:
- Opposite angles are equal: angle R = angle P = 120 degrees
- Adjacent angles are supplementary: angle Q = 180 - 120 = 60 degrees
- Opposite angles are equal: angle S = angle Q = 60 degrees
Verification: 120 + 60 + 120 + 60 = 360 degrees (angle sum property of quadrilateral).
Answer: angle P = 120 degrees, angle Q = 60 degrees, angle R = 120 degrees, angle S = 60 degrees.
Example 5: Example 5: Finding the other diagonal
Problem: The area of a rhombus is 120 cm² and one diagonal is 15 cm. Find the other diagonal.
Solution:
Given:
- Area = 120 cm²
- d₁ = 15 cm
Using the area formula:
- Area = (1/2) x d₁ x d₂
- 120 = (1/2) x 15 x d₂
- 120 = 7.5 x d₂
- d₂ = 120 / 7.5
- d₂ = 16 cm
Answer: The other diagonal is 16 cm.
Example 6: Example 6: Diagonal bisects the angle
Problem: In rhombus ABCD, diagonal AC makes an angle of 35 degrees with side AB. Find angle ABC.
Solution:
Given:
- Angle BAC = 35 degrees
In a rhombus, the diagonal bisects the vertex angle.
- So angle BAC = angle DAC = 35 degrees
- Therefore angle BAD = 35 + 35 = 70 degrees
Adjacent angles in a rhombus are supplementary:
- angle ABC = 180 - 70 = 110 degrees
Answer: angle ABC = 110 degrees.
Example 7: Example 7: Proving a quadrilateral is a rhombus
Problem: ABCD is a quadrilateral where AB = BC = CD = DA = 7 cm. Is ABCD a rhombus?
Solution:
Given:
- AB = BC = CD = DA = 7 cm (all four sides are equal)
A quadrilateral with all four sides equal is a rhombus.
We can also verify: since opposite sides are equal (AB = CD and BC = DA), ABCD is a parallelogram. Since all sides are equal, this parallelogram is a rhombus.
Answer: Yes, ABCD is a rhombus.
Example 8: Example 8: Right triangles formed by diagonals
Problem: The diagonals of a rhombus are 6 cm and 8 cm. Find the area of each right triangle formed by the diagonals.
Solution:
Given:
- d₁ = 6 cm, d₂ = 8 cm
The diagonals bisect each other at right angles, forming 4 congruent right triangles. Each right triangle has legs:
- d₁/2 = 3 cm
- d₂/2 = 4 cm
Area of each right triangle:
- = (1/2) x 3 x 4
- = 6 cm²
Verification: 4 x 6 = 24 cm². Area of rhombus = (1/2) x 6 x 8 = 24 cm². Verified.
Answer: The area of each right triangle is 6 cm².
Example 9: Example 9: Perimeter and area together
Problem: A rhombus has a perimeter of 100 cm and one diagonal of 14 cm. Find the other diagonal and the area.
Solution:
Given:
- Perimeter = 100 cm, so side = 100/4 = 25 cm
- d₁ = 14 cm, so d₁/2 = 7 cm
Finding d₂ using Pythagoras Theorem:
- side² = (d₁/2)² + (d₂/2)²
- 25² = 7² + (d₂/2)²
- 625 = 49 + (d₂/2)²
- (d₂/2)² = 576
- d₂/2 = 24
- d₂ = 48 cm
Finding the area:
- Area = (1/2) x 14 x 48 = 336 cm²
Answer: The other diagonal is 48 cm and the area is 336 cm².
Example 10: Example 10: Rhombus vs Square
Problem: A rhombus has all angles equal to 90 degrees. What special shape is it? Find the diagonals if the side is 5 cm.
Solution:
A rhombus with all angles equal to 90 degrees is a square.
Given: side = 5 cm, all angles = 90 degrees.
In a square, diagonals are equal. Let each diagonal = d.
- Using Pythagoras Theorem in the right triangle formed by the diagonal:
- d² = side² + side² = 5² + 5² = 25 + 25 = 50
- d = sqrt(50) = 5 sqrt(2) cm approx 7.07 cm
Answer: The shape is a square. Each diagonal is 5 sqrt(2) cm (approximately 7.07 cm).
Real-World Applications
The properties of a rhombus are used in many real-world situations:
- Floor Tiling: Rhombus-shaped tiles are used in decorative floor patterns. Knowing the angle and side properties helps in planning tile layouts.
- Kite Design: Traditional kites often have a rhombus shape. The diagonal properties help determine the length of the frame sticks.
- Road Signs: Diamond-shaped warning signs on roads are rhombuses. Understanding the dimensions helps in manufacturing.
- Engineering: Rhombus-shaped linkages are used in mechanical engineering — scissors jacks and expandable gates use the property that a rhombus can change its angles while keeping sides constant.
- Crystallography: Certain crystal structures have rhombus-shaped cross-sections. Scientists use rhombus properties to study mineral formations.
- Architecture: Rhombus patterns appear in window designs, wall decorations, and fence patterns in buildings.
- Optics: Rhombus-shaped prisms are used in optical instruments to redirect light beams.
Key Points to Remember
- A rhombus is a parallelogram with all four sides equal.
- Opposite angles of a rhombus are equal.
- Adjacent angles of a rhombus are supplementary (sum = 180 degrees).
- Diagonals of a rhombus bisect each other at right angles (90 degrees).
- Each diagonal of a rhombus bisects the vertex angles.
- Diagonals of a rhombus are generally NOT equal (unless it is a square).
- The diagonals divide the rhombus into 4 congruent right triangles.
- Area of rhombus = (1/2) x d₁ x d₂.
- Perimeter of rhombus = 4 x side.
- A square is a special rhombus where all angles are 90 degrees.
- A rhombus has 2 lines of symmetry (along its diagonals).
Practice Problems
- Find the perimeter of a rhombus whose side is 18 cm.
- The diagonals of a rhombus are 24 cm and 10 cm. Find its area.
- One diagonal of a rhombus is 30 cm and the side is 17 cm. Find the other diagonal.
- In a rhombus ABCD, angle A = 70 degrees. Find all other angles.
- The area of a rhombus is 150 cm² and one diagonal is 25 cm. Find the other diagonal.
- The diagonals of a rhombus are 12 cm and 16 cm. Find the side of the rhombus.
- Can a rhombus have all angles equal? If yes, what shape does it become?
- A rhombus has perimeter 52 cm and one diagonal 24 cm. Find its area.
Frequently Asked Questions
Q1. What is a rhombus?
A rhombus is a quadrilateral (four-sided figure) in which all four sides are equal in length. It is a special type of parallelogram.
Q2. Is every rhombus a parallelogram?
Yes. Every rhombus is a parallelogram because its opposite sides are equal and parallel. However, not every parallelogram is a rhombus.
Q3. Are the diagonals of a rhombus equal?
No. The diagonals of a rhombus are generally NOT equal. They are equal only when the rhombus is also a square.
Q4. At what angle do the diagonals of a rhombus bisect each other?
The diagonals of a rhombus bisect each other at right angles (90 degrees). This is a key property that distinguishes a rhombus from a general parallelogram.
Q5. How is a rhombus different from a square?
Both have all four sides equal. A square has all angles equal to 90 degrees and its diagonals are equal. A rhombus can have angles other than 90 degrees and its diagonals are generally unequal. A square is a special case of a rhombus.
Q6. How do you find the area of a rhombus?
Area of rhombus = (1/2) x d₁ x d₂, where d₁ and d₂ are the lengths of the two diagonals.
Q7. How many lines of symmetry does a rhombus have?
A rhombus has 2 lines of symmetry, both along its diagonals.
Q8. Does a diagonal of a rhombus bisect the vertex angle?
Yes. Each diagonal of a rhombus bisects the pair of vertex angles it connects. For example, diagonal AC bisects angle A and angle C.
Q9. Can all angles of a rhombus be equal?
If all four angles of a rhombus are equal, each angle must be 360/4 = 90 degrees. In that case, the rhombus becomes a square.
Q10. What is the relationship between the side and diagonals of a rhombus?
side² = (d₁/2)² + (d₂/2)². This comes from the Pythagoras theorem applied to the right triangle formed when the diagonals bisect each other at 90 degrees.
