Deriving the Perimeter of a Rectangle When Given the Side of a Square and Their Areas are Equal
Introduction
In this article, we will explore a mathematical problem involving the areas and perimeters of a square and a rectangle. We will walk through the process of solving this problem using step-by-step calculations to find the perimeter of the rectangle given the side of the square and the areas of both shapes.
Problem Statement
The perimeter of the square is 64 cm, and the areas of the square and rectangle are equal. The ratio of the length to the breadth of the rectangle is 4:1. We need to find the perimeter of the rectangle.
Step-by-Step Solution
Step 1: Calculate the Side Length of the Square
The perimeter of a square is given by the formula:
P 4s
where s is the side length of the square.
Given that the perimeter P is 64 cm, we have:
64 4s
s 64 / 4 16 , text{cm}
Step 2: Calculate the Area of the Square
The area of the square is given by the formula:
A s^2
Using the side length calculated in Step 1:
A 16^2 256 , text{sq cm}
Step 3: Set Up the Dimensions of the Rectangle
Let the length of the rectangle be 4x cm and the breadth be x cm. Given the ratio of length to breadth is 4:1, the area of the rectangle can be expressed as:
Area 4x times x 4x^2 256 , text{sq cm}
Solving for x:
x^2 256 / 4 64
x sqrt{64} 8 , text{cm}
So, the breadth of the rectangle is x 8 , text{cm}
And the length of the rectangle is:
4x 4 times 8 32 , text{cm}
Step 4: Calculate the Perimeter of the Rectangle
The perimeter of the rectangle is given by the formula:
P_r 2(l b)
where l is the length and b is the breadth of the rectangle.
Substituting the dimensions found in the previous steps:
P_r 2(32 8) 2 times 40 80 , text{cm}
Conclusion
The perimeter of the rectangle is 80 , text{cm}.