Exploring the Ratio of Areas: Equal Perimeters of Rectangles and Squares

Understanding the relationship between the perimeter and area of geometric shapes is a fundamental concept in mathematics. In this article, we will delve into a problem where a rectangle and a square have equal perimeters, and the ratio of their adjacent sides is 1:2. We will calculate and compare their areas to determine the ratio of these areas. This exploration will be useful for students and educators alike, as it ties together concepts from geometry and algebra.

Introduction

Let's denote the dimensions of the rectangle as length (l) and width (w). Given that the ratio of the adjacent sides of the rectangle is 1:2, we can express the dimensions as:

(l 2x, w x)

for some positive value (x). Here, (x) represents the smaller side of the rectangle, and by extension, the width of the rectangle.

Step 1: Calculate the Perimeter of the Rectangle

The perimeter (P) of the rectangle is given by:

(P 2l 2w 2(2x) 2x 4x 2x 6x)

Step 2: Calculate the Perimeter of the Square

Let the side length of the square be (s). The perimeter (P) of the square is:

(P 4s)

Step 3: Set the Perimeters Equal

Since the perimeters of the rectangle and the square are equal, we have:

(6x 4s)

From this, we can express (s) in terms of (x) as:

(s frac{6x}{4} frac{3x}{2})

Step 4: Calculate the Areas

The area (A_r) of the rectangle is:

(A_r l times w 2x times x 2x^2)

The area (A_s) of the square is:

(A_s s^2 left(frac{3x}{2}right)^2 frac{9x^2}{4})

Step 5: Calculate the Ratio of the Areas

Now we find the ratio of the area of the rectangle to the area of the square:

(text{Ratio} frac{A_r}{A_s} frac{2x^2}{frac{9x^2}{4}} frac{2x^2 times 4}{9x^2} frac{8}{9})

Thus, the ratio of the areas of the rectangle to the square is:

(8:9)

Wrapping up the explanation, the area calculation of the rectangle and the square leads us to the ratio (8:9). This method can be applied to other similar problems involving rectangles and squares with equal perimeters, and it reinforces the interplay between dimensions, perimeters, and areas in geometric shapes.

Thus, the final answer is:

(boxed{frac{8}{9}})