Introduction

Determining the side length of a square when given its diagonal is a common problem in geometry. This article outlines the process with various methods and explains the application of the Pythagorean Theorem in solving such problems. We'll also discuss the reasoning behind each step and how to verify the final answer using different calculations.

Understanding the Geometry of Squares

A square is a regular quadrilateral with all sides equal and all internal angles 90 degrees. The diagonal of a square forms a right triangle with two adjacent sides. To find the side length, we can use the Pythagorean Theorem, which states that the square of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides.

Method 1: Using the Direct Formula

Let's denote the side of the square as S and the diagonal as L. The relationship between the side and the diagonal of a square is given by:

[ L Ssqrt{2} ]

Rearranging to solve for S:

[ S frac{L}{sqrt{2}} ]

Given that the diagonal L is 42 - √2, we substitute this value into the formula:

[ S frac{42 - sqrt{2}}{sqrt{2}} ]

Simplifying further:

[ S frac{42 - sqrt{2}}{sqrt{2}} frac{42}{sqrt{2}} - frac{sqrt{2}}{sqrt{2}} 21sqrt{2} - 1 ]

Since √2 ≈ 1.414, we can approximate:

[ S 21 times 1.414 - 1 29.698 - 1 28.698 units

Method 2: Using Pythagorean Theorem

Let's denote the side of the square as S. The diagonal L is the hypotenuse of the right-angled triangle formed by the two sides.

[ L^2 2S^2 ]

Rearranging for S gives:

[ S^2 frac{L^2}{2} ]

Given that L 42 - √2, we square both sides:

[ (42 - √2)^2 2S^2 ]

Simplifying the left side:

[ 42^2 - 2 times 42 times √2 (√2)^2 2S^2 ]

42^2 - 84√2 2 2S^2.

[ 1764 - 84√2 2 2S^2 ]

Combining terms:

[ 1766 - 84√2 2S^2 ]

Solving for S:

[ S sqrt{frac{1766 - 84√2}{2}} ]

Approximating √(1766 - 84√2)/√2:

[ S sqrt{frac{1766 - 84√2}{2}} 28.698 units ]

Comparative Analysis and Verification

The two methods yield nearly identical results. Method 1 uses a simpler formula, while Method 2 involves a more detailed step-by-step calculation. Both approaches validate that the side length of the square is approximately 28.698 units.

Conclusion

Whether using direct formulas or the Pythagorean Theorem, determining the side length of a square from its diagonal involves careful application of geometric principles. The methods outlined here provide a robust framework for solving similar problems in geometry and help reinforce the underlying principles of square and diagonal relationships.