Solving for Rectangle Dimensions: A Comprehensive Guide

Understanding and calculating the dimensions of a rectangle based on given measurements is a fundamental concept in geometry. This guide provides step-by-step instructions, including the use of formulas, for determining the dimensions based on the perimeter and a given relationship between the length and width. By following these methods, you can accurately find the dimensions of a rectangle.

Problem Statement

Given a rectangle with a perimeter of 108 cm, where the length is 25% larger than the width, we aim to determine the exact dimensions of the rectangle.

Step-by-Step Solution

Using the Perimeter Formula

The perimeter P of a rectangle is given by the formula:

P 2(length width)

In this case, P 108 cm. We also know that the length L is 25% larger than the width W. This can be expressed as:

L W 0.25W 1.25W

Substituting and Simplifying

Substitute the expression for L into the perimeter formula:

108 2(1.25W W)

Combine like terms:

108 2(2.25W) or 108 4.5W

Solving for Width and Length

To find the width W, divide both sides by 4.5:

W 108 / 4.5 24 cm

To find the length L:

L 1.25W 1.25 times 24 30 cm

Alternative Methods

There are several other methods to solve this problem. Let's look at a few additional approaches:

Direct Substitution

Given:

L B 25B/100 125B/100 5B/4

And:

2LB 108

Substitute the value of L into the formula:

2 times 9B/4 times B 108

9B/2 108

Solve for B (width):

B 108 times 2 / 9 24 cm

Calculate the length:

L 5 times 24 / 4 30 cm

Number Line Method

A simpler approach involves breaking down the problem into smaller steps:

108 - 25 times 2 48

48 / 4 12

So, the width W 12 cm

Now, calculate the length:

L 12 25 37 cm

Verify using the perimeter formula:

2(37 12) 108

Verification

To ensure accuracy, we can recheck the dimensions:

Perimeter 2(30 24) 2 times 54 108 cm

The dimensions are correct.

Conclusion

In summary, by using the perimeter formula and the percentage relationship between the length and width, we can determine that the width of the rectangle is 24 cm and the length is 30 cm. Understanding and mastering these concepts can significantly enhance your problem-solving skills in geometry.