1. Definition of a Rectangle
A rectangle is a special type of quadrilateral in which all four angles are right angles (each measuring \(90^\circ\)).
It also has two pairs of equal and parallel opposite sides. We usually name it as \(ABCD\) with \(AB = CD\) and \(BC = AD\).
Although it looks like a box or book shape, its geometric properties make many problems easier to solve.
2. Properties of a Rectangle
Rectangles are among the most important quadrilaterals because they combine properties of both parallelograms and right-angled figures.
2.1. Opposite Sides are Equal and Parallel
In rectangle \(ABCD\):
\(AB = CD \quad \text{and} \quad BC = AD\)
Both pairs of opposite sides are parallel.
2.2. All Angles are Right Angles
Each interior angle is:
\(\angle A = \angle B = \angle C = \angle D = 90^\circ\)
This is the key feature of a rectangle.
2.3. Diagonals are Equal
The diagonals of a rectangle have equal length:
\(AC = BD\)
This helps in many coordinate geometry and construction problems.
2.4. Diagonals Bisect Each Other
Just like parallelograms, the diagonals of a rectangle cut each other into two equal halves:
\(AO = OC\) and \(BO = OD\)
where \(O\) is the point of intersection.
3. Perimeter and Area of a Rectangle
A rectangle's formulas are simple and frequently used in daily-life applications like flooring, painting, fencing, and design.
3.1. Perimeter
If length = \(l\) and breadth = \(b\), then:
\(\text{Perimeter} = 2(l + b)\)
3.2. Area
The area formula is:
\(\text{Area} = l \times b\)
3.3. Quick Example
A rectangle with length \(12\text{ cm}\) and breadth \(7\text{ cm}\) has area:
\(12 \times 7 = 84\text{ cm}^2\)
4. Diagonals of a Rectangle
The diagonals of a rectangle are equal and can be found using the Pythagorean theorem.
4.1. Diagonal Length Formula
If the length is \(l\) and breadth is \(b\), then the diagonal is:
\(d = \sqrt{l^2 + b^2}\)
4.2. Example
If \(l = 9\) cm and \(b = 12\) cm:
\(d = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15\text{ cm}\)
5. Rectangle as a Special Parallelogram
A rectangle is essentially a parallelogram with all angles equal to \(90^\circ\). So it inherits all parallelogram properties, with extra advantages because of its right angles.
5.1. Inherited Properties
- Opposite sides are equal.
- Opposite sides are parallel.
- Diagonals bisect each other.
5.2. Additional Features
- All four angles are right angles.
- Diagonals are equal.