# Properties of a Rectangle Rhombus and Square | Special Parallelograms Properties

Properties of a Rectangle Rhombus and Square is always a confusing concept for students. Learning every individual topic is important to score good marks in the exam. So, we have explained every individual topic clearly in a detailed manner in this article. Therefore, those who wish to learn the concepts of Parallelogram and its properties, problems, can completely learn the Parallelogram concepts on our website.

### Rectangle

A rectangle is said to be a parallelogram when it has all 4 angles having equal measure.

#### Properties of Rectangle

• The Opposite sides of a rectangle are parallel.
• Also, the Opposite sides of a rectangle are equal in length.
• Diagonals are equal in length.
• The interior angles are 90 degrees each.
• Diagonals bisect each other.
• It has horizontal and vertical lines of symmetry.
• Each of the diagonal bisects the rectangle into 2 congruent triangles.
• If you combine the 4 sides of a rectangle, then the mid-points of it form a rhombus.

#### Rectangle Formulas

If l is the length of the rectangle and b is the breadth of the rectangle, then
Area = lb square units
Perimeter = 2 (l+b) units.

#### Diagonal Properties of a Rectangle

Prove that the diagonals of a rectangle are equal and bisect each other.

Proof:
Let PQRS be a rectangle that has diagonals PQ and QS intersect at the point O. From ∆ PQR and ∆ QPS,
PQ = QP (common)
∠PQR = ∠QPS (each equal to 90º)
QR = PS (opposite sides of a rectangle).
Therefore, ∆ PQR ≅ ∆ QPS (by SAS congruence)
⇒ PR = QS.
Hence, the diagonals of a rectangle are equal.
From ∆ OPQ and ∆ ORS,
∠OPQ = ∠ORS (alternate angles)
∠OQP = ∠OSR (alternate angles)
PQ = RS (opposite sides of a rectangle)
Therefore, ∆OPQ ≅ ∆ ORS. (by ASA congruence)
⇒ OP = OR and OQ = OS.
This shows that the diagonals of a rectangle bisect each other.

Hence, the diagonals of a rectangle are equal and bisect each other.

### Rhombus

The rhombus is a quadrilateral that consists of four sides with equal lengths.

#### Properties of Rhombus

• The Rhombus consists of parallel and equal opposite sides. As it consists of parallel and equal opposite sides, it is said to be a parallelogram.
• All available sides (4 sides) are equal.
• Also, opposite angles in a rhombus are equal.
• Diagonals bisect each other.
• Diagonals of a rhombus intersect each other at right angles.
• Furthermore, Diagonals bisect opposite vertex angles.
• Every diagonal divides the rhombus into 2 congruent triangles.

#### Rhombus Formula

If b is the side, a and b are the two diagonals of the rhombus, then
Area = ab/2 Square units.
Perimeter = 4b units

#### Diagonal Properties of a Rhombus

Prove that the diagonals of a rhombus bisect each other at right angles.

Proof:
Let PQRS be a rhombus whose diagonals AC and BD intersect at point O. The diagonals of a parallelogram bisect each other. Also, we know that every rhombus is a parallelogram.
So, the diagonals of a rhombus bisect each other.
Therefore, OP = OR and OQ = OS
From ∆ ROQ and ∆ ROS,
RQ = RS (sides of a rhombus)
RO = RO (common).
OQ = OS (proved)
Therefore, ∆ ROQ ≅ ∆ ROS (by SSS congruence)
⇒ ∠ROQ = ∠ROS
But, ∠ROQ + ∠ROS = 2 right angles (linear pair)
Therefore, ∠ROQ = ∠ROS = 1 right angle.

Hence, the diagonals of a rhombus bisect each other at right angles.

### Square

A square is a rectangle that has all equal sides.

#### Properties of Square

• The opposite sides of a square are parallel.
• All 4 sides are equal in length.
• Diagonals are equal in length.
• Diagonals bisect opposite vertex angles.
• The interior angles of a square measure 90 degrees each.
• Diagonals bisect each other at right angles.
• It has 4 lines of symmetry – a horizontal, a vertical, and 2 diagonals.
• Each diagonal bisects the square into 2 congruent triangles.

#### Square Formula

If b is the side of the square, then
Area = b² square units
Perimeter = 4b units.

#### Diagonal Properties of a Square

Prove that the diagonals of a square are equal and bisect each other at right angles.

Proof:
We know that the diagonals of a rectangle are equal.
Also, every square is a rectangle.
Therefore, the diagonals of a square are equal.
Again, the diagonals of a rhombus bisect each other at right angles. But, every square is a rhombus.
So, the diagonals of a square bisect each other at right angles.

Hence, the diagonals of a square are equal and also bisect each other at right angles.

Note 1: If the diagonals of a quadrilateral are equal but it is not necessary to be a rectangle.
Note 2: If the diagonals of a quadrilateral interest at a point with right angles then also it is not necessary to become a rhombus.