In Rectangle ABCD, Point E Divides Side BC Such That BE:EC = 2:1, And Point F Divides Side DC Such That DF:FC = 2:3. If The Area Of Rectangle ABCD Is 45 Square Units, What Is The Area Of Quadrilateral AECF?

by ADMIN 207 views

Introduction

In the realm of geometry, understanding the relationships between areas of different shapes within a larger figure is a fundamental skill. This article delves into a specific problem involving a rectangle ABCD, where points E and F divide sides BC and DC in given ratios. Our main goal is to determine the area of quadrilateral AECF, given that the area of rectangle ABCD is 45 square units. Let's break down the problem step by step, using geometric principles and area calculations to arrive at the solution. This exploration will not only provide the answer but also enhance our understanding of spatial reasoning and problem-solving techniques in geometry. We will explore how ratios, areas, and geometric shapes interplay to solve complex problems, making this a valuable exercise for anyone interested in mathematics and spatial puzzles.

Problem Statement

Let's clearly state the problem we're about to solve. In rectangle ABCD, point E divides side BC in the ratio 2:1, and point F divides side DC in the ratio 2:3. The area of rectangle ABCD is 45 square units. We need to find the area of quadrilateral AECF. This problem combines concepts of ratios, areas, and geometric shapes, requiring a methodical approach to solve it accurately. Understanding the problem statement is the first crucial step in any mathematical endeavor. We must visualize the rectangle, the points E and F, and the quadrilateral AECF to effectively tackle the problem. Breaking down the information given—the ratios, the area, and the shape—will guide us in formulating a plan for the solution. The ultimate goal is to calculate the area of AECF, which will involve some strategic calculations and geometric insights.

Setting Up the Solution

To solve this problem, we'll first establish some notations and relationships. Let AB = CD = l (length) and BC = AD = b (breadth) of the rectangle ABCD. Since the area of rectangle ABCD is 45 sq. units, we have l * b* = 45. Point E divides BC in the ratio 2:1, so let BE = (2/3)b and EC = (1/3)b. Similarly, point F divides DC in the ratio 2:3, so let DF = (2/5)l and FC = (3/5)l. Now, the area of quadrilateral AECF can be found by subtracting the areas of triangles ABE and ADF from the area of rectangle ABCD. This is a strategic approach as it breaks down the complex shape into simpler components. Visualizing these divisions and relationships is crucial for understanding the problem thoroughly. We are essentially using the principle of area subtraction to find the area of the desired quadrilateral. By setting up these notations and relationships, we create a framework for our calculations, making the solution process clearer and more organized.

Calculating Areas

Now, let's calculate the areas of triangles ABE and ADF. The area of triangle ABE is (1/2) * AB * BE = (1/2) * l * (2/3)b = (1/3) * l * b*. The area of triangle ADF is (1/2) * AD * DF = (1/2) * b * (2/5)l = (1/5) * l * b*. Since we know l * b* = 45, the area of triangle ABE is (1/3) * 45 = 15 sq. units, and the area of triangle ADF is (1/5) * 45 = 9 sq. units. These calculations are straightforward applications of the formula for the area of a triangle, which is half the base times the height. It's essential to accurately substitute the values and simplify the expressions. By breaking down the areas into these smaller components, we make the overall problem more manageable. The numerical results for the areas of triangles ABE and ADF are critical for the next step, where we will subtract these areas from the total area of the rectangle to find the area of quadrilateral AECF.

Finding the Area of Quadrilateral AECF

Now we subtract the areas of triangles ABE and ADF from the area of rectangle ABCD to find the area of quadrilateral AECF. The area of AECF = Area of ABCD - (Area of ABE + Area of ADF). Substituting the values we calculated, Area of AECF = 45 - (15 + 9) = 45 - 24 = 21 sq. units. Therefore, the area of quadrilateral AECF is 21 square units. This final step combines the results of our previous calculations to arrive at the solution. It showcases the power of breaking down a complex problem into smaller, manageable parts. By carefully computing the areas of the triangles and subtracting them from the total area, we have successfully found the area of the quadrilateral. This solution demonstrates a methodical approach to geometric problems, where understanding the relationships between shapes and areas is key to finding the answer. The calculated area, 21 sq. units, is the solution we were seeking, marking the successful completion of the problem.

Final Answer and Conclusion

The area of quadrilateral AECF is 21 square units. Thus, the correct answer is (c) 21 sq. units. This problem demonstrates the application of geometric principles, ratios, and area calculations to solve a complex problem. By breaking down the rectangle into smaller shapes and using the given ratios, we were able to systematically find the area of the quadrilateral. This type of problem is not only a great exercise in geometry but also enhances our problem-solving skills in general. Understanding the underlying concepts and applying them methodically is crucial for success in mathematics and beyond. The solution highlights the importance of visualization, strategic planning, and accurate calculations. With the final answer in hand, we can appreciate the elegance of geometric problem-solving and the power of breaking down complex shapes into simpler components. This exercise serves as a valuable lesson in mathematical reasoning and spatial understanding.

Repair Input Keyword

In rectangle ABCD, point E divides side BC such that BE:EC = 2:1, and point F divides side DC such that DF:FC = 2:3. If the area of rectangle ABCD is 45 square units, what is the area of quadrilateral AECF?

SEO Title

Area of Quadrilateral in a Rectangle Step-by-Step Solution