Folding a Rectangle Along Its Diagonal: Area Calculation and Geometry

When a rectangle is folded along its diagonal, the resulting figure often forms a triangle. This article will explore the geometry and the calculation of the area of this figure, providing both the simpler and more detailed approaches.

Introduction

A rectangle is a quadrilateral with four right angles. The given problem involves a rectangle with a length of 18 cm and a breadth of 12 cm. When it is folded along its diagonal, it results in a triangle whose area can be calculated using basic geometric principles.

Calculation of the Area of a Triangle Formed by Folding a Rectangle

When a rectangle is folded along its diagonal, the resulting figure is a triangle. The area of this triangle can be found using the formula for the area of a triangle:

Area 0.5 times; base times; height

In this specific scenario, the base is the length of the rectangle, which is 18 cm, and the height is the breadth of the rectangle, which is 12 cm. Let's calculate the area:

Area  0.5 times; 18 cm times; 12 cm      0.5 times; 216 cm2      108 cm2

Therefore, the area of the triangle formed by folding the rectangle along its diagonal is 108 cm2.

Explanation of the Geometry

Consider a rectangle ABCD with length 18 cm and breadth 12 cm. The diagonal divides the rectangle into two congruent right-angled triangles. The area of the original rectangle ABCD is calculated as:

Area  length times; breadth      18 cm times; 12 cm      216 cm2

Since the diagonal divides the rectangle into two equal areas, the area of the triangle formed by folding along the diagonal is half of the area of the rectangle:

Area  216 cm2 / 2      108 cm2

Detailed Calculation of the Area

For a more detailed calculation, let's consider a general rectangle with sides (a) and (b). The area of the rectangle is (a times b). When the rectangle is folded along its diagonal, the resulting figure is a triangle whose area can be found by subtracting the area of an overlapping region from the area of the rectangle.

The base of the overlapping region (triangle BCE) is the diagonal of the rectangle, which is (sqrt{a^2 b^2}). The height of the triangle can be found using similar triangles BEF and BCA. Since F is the midpoint of BC, we have:

EF / FB  CA / AB

Given that (EF frac{sqrt{a^2 b^2}}{2}) and (CA a), (AB b), we get:

EF / (sqrt{a^2   b^2} / 2)  a / bEF  (sqrt{a^2   b^2} times; a) / 2b

The area of the overlapping region (triangle BCE) is:

Area of BCE  0.5 times; (sqrt{a^2   b^2} times; a / 2b) times; sqrt{a^2   b^2}             (a times; (a^2   b^2)) / (4b)

The required area of the figure formed is the area of the rectangle minus the area of the overlapping region:

Required Area  a times; b - (a times; (a^2   b^2)) / (4b)               (4ab^2 - a^3 - b^3) / (4b)

For the specific case where (a 12) cm and (b 18) cm:

Required Area  (4 times; 12 times; 18^2 - 12^3 - 18^3) / (4 times; 18)               138 cm2

Hence, the area of the figure formed is 138 cm2.

Conclusion

Folding a rectangle along its diagonal is a fascinating geometric problem that involves understanding the properties of triangles and the relationship between the rectangle and its diagonal. The area of the resulting triangle can be calculated using straightforward geometric principles or more detailed algebraic methods. Understanding these processes can help in solving similar problems in geometry and other related fields.

For more insights into the concepts discussed, consider exploring the topics of similar triangles, right triangles, and the properties of rectangles and their diagonals. These concepts are fundamental in geometry and have numerous practical applications in various fields such as architecture, engineering, and design.