Properties Of Rectangles

A rectangle is one of the most common and basic shapes of geometry having applications from academic curriculum to the larger works of architectural designs. Hence, it is crucial to understand the properties of rectangles for solving various geometric and real-life problems. This article will help you explore the basic properties of this simple yet important figure of geometry.

1.0Definition Of Rectangle

The rectangle is a two-dimensional Quadrilateral identified by four sides crossing each other at a right angle (90°) with equal and opposite sides. It is a special case of a parallelogram. Mathematically, if a rectangle holds vertices, say A, B, C, and D, then these are the conditions it will follow: 

• ∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°
• AB = CD and AD = BC
• The diagonals AC and BD are congruent (which is AC = BD).

2.0Properties of Rectangle

The properties of rectangles are used in solving geometric problems, which include congruency and similarity questions related to different geometric figures. The basic properties of a rectangle with a diagram, as shown above, are: 

• Rectangle in Geometry: As mentioned earlier Rectangle is a quadrilateral, a special type of parallelogram that possesses the properties of a parallelogram, too. 
• Sides of the Rectangle: Opposite sides are equal and parallel to each other and can be mentioned as AB ∥ CD & AD ∥ BC. 
• Diagonals of Rectangle: 
• • The diagonals of a rectangle are congruent, meaning they are equal to one another in length. 
  • Diagonals of the rectangle bisect each other, or in simple words, they intersect each other at their midpoint and get divided into two equal halves. 
  • A diagonal divides the rectangle into two congruent right triangles.

• The sum of Angles: As the measure of each angle of the rectangle is 90°, The sum of adjacent as well as opposite angles of the rectangle is 180° and the sum of all the angles of a rectangle is 360°.
• Symmetry: A rectangle has two lines of reflectional symmetry, a property in which one half of an object is the mirror image of another half. The Rectangle has mirror symmetry, one through the midpoints of opposite sides – vertical symmetry and another through the other set of opposite sides - horizontal symmetry. 

• Circumscribed Circle: 
• • A rectangle can be inscribed in a circle. In other words, a circle can pass through all four vertices of the rectangle. 
  • The diagonals of the rectangle are the diameters of the circumscribed circle.

• The special case of Rectangle: Rectangles with all sides and both diagonals equal are classified as squares. Hence, we can say not all rectangles are squares. 

3.0Formulas Related to the Rectangle

Rectangles, with their straightforward structure, offer simple yet important calculative properties used to measure the area, perimeter, and length of their diagonal. These formulas include: 

1. Area of Rectangle: The total space contained by the rectangle is known as the area of a rectangle. It represents the measure of the coverage of the surface area by the rectangle and is measured in square units. The formula to find the area of a rectangle with length (l) & breadth (b) is: 

$\text{Area of rectangle(A)}=l\times b$

2. The perimeter of a rectangle: The perimeter of any given rectangle is the distance around the rectangle. It's the distance you would walk if you walked around the edges of the rectangle. The formula of the perimeter is: 

$\text{Perimeter of a rectangle}=2(l+b)$

3. The diagonal of a Rectangle: The diagonal of a rectangle is the straight line connecting opposite corners. It is an important concept in geometry and has several practical applications, such as finding the shortest distance across a rectangular space.

$\text{Diagonal of rectangle(d)}=\sqrt{l^2+b^2}$

4.0Properties of Rectangle Examples

Problem 1: Find the area and diagonal of a rectangle with a length 12 cm and a width 5 cm.

Solution:

Given that length = 12cm, and width = 5cm

$\text{Area of rectangle(A)}=l\times b$

$A=12\times 5$

$A=60c{m}^2$

Problem 2: Prove that opposite triangles, formed by the intersection of two diagonals of a rectangle are congruent. 

To prove: AOD ⩭ BOC

Given: ABCD is a rectangle

Solution: AOD and BOC

AO = OC (Diagonals of a rectangle bisect each other) 

DO = BO (Diagonals of a rectangle bisect each other)

AD =  BC (Opposite sides of a rectangle) 

AOD ⩭ BOC (SSS) 

Problem 3: A rectangle has a diagonal of length 10 cm, and one of its sides is 6 cm. Find the length of the other side and calculate the area of the rectangle.

Solution:

Given that diagonal (d) = 10cm, and let l = 6, b =?

$\text{Diagonal of rectangle(d)}=\sqrt{l^2+b^2}$

$10=\sqrt{6^2+b^2}$

Squaring both sides, 

$100=36+b^2$

$b^2=100-36=64$

b=8cm