Exploring the Dimensions of a Rectangle Given Perimeter and Length-Width Relationship

When working with geometric shapes, particularly rectangles, understanding how to use given conditions to determine their dimensions can help solve various mathematical problems. This article delves into the process of finding the dimensions of a rectangle when provided with the perimeter and the relationship between the length and width. Understanding these processes can be crucial for learners and professionals alike.

Understanding the Given Problem

In this case, we are given that the perimeter of a rectangle is 78 cm and the length is 7 cm more than the width. Let’s break down the steps to solve the problem:

Step-by-Step Solution

Define Variables: Let the width be w cm. Then the length l can be expressed as:

l  w   7

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

P  2l   2w

Substitute the Given Perimeter: We know that the perimeter is 78 cm, so we set up the equation:

2l   2w  78

Substitute the Expression for Length: Substitute l w 7 into the perimeter equation:

2(w   7)   2w  78

Simplify and Solve for Width:

2w   14   2w  784w   14  784w  64w  16

The width of the rectangle is 16 cm.

Calculate the Length:

l  w   7  16   7  23

The length of the rectangle is 23 cm.

Therefore, the dimensions of the rectangle are:

Width: 16 cm Length: 23 cm

Additional Explorations

For additional practice, let’s consider another example where half of the perimeter is given. In this case, let’s assume:

Another Example

Given: Half of the perimeter is 40 inches, and the length is 6 inches more than the width.

Steps:

Width Calculation: Half of the perimeter is 40 inches, which means the full perimeter is 80 inches. The width and length relationship is given as: w (w 6) 40. Solving for width:

2w   6  402w  34w  17

The width of the rectangle is 17 inches.

Length Calculation:

l  w   6  17   6  23

The length of the rectangle is 23 inches.

Therefore, the dimensions of the rectangle are:

Width: 17 inches Length: 23 inches

Calculation of Area and Diagonal

Area can be calculated as:

A  17 inches ? 23 inches  391 inches^2

Diagonal can be calculated using the Pythagorean theorem:

D  sqrt{17^2   23^2}  sqrt{818} ≈ 28.6 inches

Conclusion

Understanding the relationship between the length and width of a rectangle, along with the perimeter, allows us to determine the dimensions accurately. This knowledge can be applied in various fields, including architecture, engineering, and mathematics. Whether dealing with the dimensions of a rectangle, a square, or a more complex geometrical figure, mastering these concepts is fundamental.