Introduction

The Pythagorean Theorem is a cornerstone of geometry and has broad applications from basic mathematics to complex problem-solving. One such application involves finding the length of a rectangle given its width and the fact that the diagonal exceeds the length by a certain amount. In this article, we'll walk through a step-by-step process to solve such a problem using the Pythagorean Theorem, focusing on the relationship between the diagonal, the width, and the length.

Understanding the Problem

Let's start with a problem statement: the diagonal of a rectangle exceeds its length by 2 cm, and the width of the rectangle is 10 cm. The task is to find the length of the rectangle.

Solving the Problem Using the Pythagorean Theorem

The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides (the width and the length).

Step 1: Define the Variables

Let the length of the rectangle be ( l ) cm. The width of the rectangle is given as ( w 10 ) cm. The diagonal of the rectangle is ( d l 2 ) cm.

Step 2: Apply the Pythagorean Theorem

According to the Pythagorean Theorem:

[ d^2 l^2 w^2 ]

Substituting the known values and the expression for the diagonal:

[ (l 2)^2 l^2 10^2 ]

Step 3: Expand and Simplify the Equation

Expanding the left side of the equation:

[ l^2 4l 4 l^2 100 ]

Subtract ( l^2 ) from both sides:

[ 4l 4 100 ]

Subtract 4 from both sides:

[ 4l 96 ]

Divide both sides by 4:

[ l 24 ]

Hence, the length of the rectangle is 24 cm.

Verification

To verify our solution, let's substitute the values back into the equation:

[ (24 2)^2 24^2 10^2 ] [ 26^2 576 100 ] [ 676 676 ]

The equation holds true, confirming that our solution is correct.

Conclusion

In this article, we explored how to find the length of a rectangle when the diagonal exceeds the length by a given amount, using the Pythagorean Theorem. By applying the theorem, we were able to derive and solve the equation accurately. This method not only helps in solving specific problems but also reinforces the foundational geometric principles.

Additional Resources

If you're interested in learning more about the Pythagorean Theorem and its applications, you might also want to explore:

Basic Geometry: Understanding geometry concepts and theorems. Algebraic Solutions: Solving equations using algebraic methods. Geometry Problems: Practice problems to improve your skills.