Understanding Perimeter and Ratio in Rectangles

Rectangles are a common geometric shape in various applications, from simple arithmetic problems to real-world scenarios such as home renovation and landscaping. In this article, we will explore the relationship between the perimeter, length, and breadth of a rectangle, specifically through a problem that requires understanding both the perimeter and the ratio of its dimensions. We will investigate the steps and reasoning behind solving such problems to ensure a thorough understanding of the underlying concepts.

The Problem: A Rectangle's Perimeter and Ratio

The problem at hand involves a rectangle with a given perimeter of 64 cm, and the length and breadth are in the ratio of 3:1. To solve this problem, we need to find the exact dimensions of the rectangle, which are its length and breadth. Let's break down the steps to solve this problem accurately.

Step-by-Step Solution

1. **Understanding the Given Information**: The perimeter of the rectangle is 64 cm, and the ratio of the length to the breadth is 3:1. This means if the length is 3x, then the breadth is x.

2. **Formulating the Equation for Perimeter**: The perimeter of a rectangle is given by the formula 2(Length Breadth). Using the given ratio, we can express the perimeter in terms of x:

```math 2(3x x) 64 ```

3. **Solving for x**: Simplify and solve for x:

```math 2(4x) 64 8x 64 x 8 ```

4. **Finding the Length and Breadth**: With x found, we can now determine the actual values for the length and breadth:

```math Length 3x 3(8) 24 cm Breadth x 8 cm ```

Therefore, the length of the rectangle is 24 cm, and the breadth is 8 cm.

Reviewing the Problem

It's essential to verify the solution and ensure that all units are correct. For instance, the perimeter is measured in linear units (cm), not in square units (cm2). In the original problem, there was an error that misstated the perimeter as 642, which is incorrect. The correct perimeter is 64 cm.

Final Answer

Based on the steps above, the length and breadth of the rectangle are:

Length 24 cm

Breadth 8 cm

Conclusion

Understanding the relationship between the perimeter and the ratio of dimensions in rectangles is crucial for solving various geometric problems. By breaking down the problem into manageable steps and ensuring the correct application of formulas, we can accurately determine the dimensions of a rectangle. This knowledge is not only valuable for mathematical problems but also for practical applications such as layout design and construction.

Additional Tips

Always double-check units when dealing with perimeter and area to avoid common mistakes. Use the ratio to set up equations that represent the given relationships. Verify your solution by plugging the values back into the original problem. Practice similar problems to reinforce your understanding of geometric properties.

By following these steps and tips, you can confidently solve problems related to the perimeter and ratio of rectangles in various mathematical and real-world contexts.