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.