Solving for the Dimensions of a Rectangle with Given Diagonal and Area

Understanding the relationship between the length, width, diagonal, and area of a rectangle can be crucial in various mathematical and real-world applications. This article will guide you through the process of solving for the dimensions of a rectangle given its diagonal and area, using the Pythagorean theorem and basic algebraic principles. We will also explore real-world applications and practical scenarios where these concepts are applicable.

Problem Statement

We are given a rectangle with a diagonal of 25 cm and an area of 168 cm2. The task is to determine the possible length and width of the rectangle.

Methodology

To solve this problem, we will use the following two fundamental formulas:

The area A of a rectangle is given by: (A l times w) The diagonal d of a rectangle can be calculated using the Pythagorean theorem: (d sqrt{l^2 w^2})

Step-by-Step Solution

Given: Area (A 168 , text{cm}^2) Diagonal (d 25 , text{cm})

We can set up the equations based on the given information:

(l times w 168) (l^2 w^2 25^2 625)

Express (w) from the first equation:

(w frac{168}{l})

Substitute (w) into the second equation:

(l^2 left(frac{168}{l}right)^2 625)

Multiply through by (l^2) to eliminate the fraction:

(l^4 - 625l^2 28224 0)

Let (x l^2), then we have:

(x^2 - 625x 28224 0)

Use the quadratic formula to solve for (x):

(x frac{625 pm sqrt{625^2 - 4 cdot 1 cdot 28224}}{2})

Calculate the discriminant:

(sqrt{625^2 - 4 cdot 1 cdot 28224} sqrt{390625 - 112896} sqrt{277729} 527)

Substitute back into the quadratic formula:

(x frac{625 pm 527}{2})

Calculate the two possible values for (x):

(x frac{1152}{2} 576) → (l sqrt{576} 24 , text{cm}) (x frac{98}{2} 49) → (l sqrt{49} 7 , text{cm})

Conclusion

Thus, the possible dimensions for the rectangle are:

Length (l 24 , text{cm}) and Width (w frac{168}{24} 7 , text{cm}) Length (l 7 , text{cm}) and Width (w frac{168}{7} 24 , text{cm})

Therefore, the length of the rectangle can be either 24 cm or 7 cm.

Real-World Applications

This method of solving for the dimensions of a rectangle can be applied in various fields such as architecture, engineering, and design. For instance, when designing a rectangular room or a rectangular garden, it is important to ensure that the dimensions meet the requirements for both the area and the diagonal, providing a comfortable and functional space.

Additional Resources

Explore more on the geometry and algebraic concepts related to rectangles and their properties. Practice with our interactive problem sets and solutions. Feedback on our website can be provided on our contact page.

Keywords: rectangle, diagonal, area