MathLAB Journal

Exploring a New Proof for the Pythagorean Theorem

Received: 02-Sep-2024, Manuscript No. mathlab-24-147745; Editor assigned: 04-Sep-2024, Pre QC No. mathlab-24-147745 (PQ); Reviewed: 18-Sep-2024, QC No. mathlab-24-147745; Revised: 23-Sep-2024, Manuscript No. mathlab-24-147745 (R); Published: 30-Sep-2024

Description

The Pythagorean Theorem, one of the cornerstones of geometry, states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is simple yet profound relationship has been known since antiquity, and the countless proofs have been devised over here for the centuries. While many traditional proofs rely on geometric constructions such as rearranging squares and triangles to visually demonstrate the relationship modern approaches to proving the theorem often involve algebra, coordinate geometry, or even trigonometry. Some new proofs might involve advanced techniques like vector analysis or complex numbers, highlighting the theorem’s deep connection to different areas of mathematics. Exploring new proofs not only reinforces the theorem’s validity but also deepens our understanding of mathematical structures. For example, by using transformations such as rotations or reflections, new insights emerge about the symmetry of space. Algebraic proofs may use properties of similar triangles or the relationships between ratios, showing how multiple perspectives converge to the same truth. Each new proof of the Pythagorean Theorem reflects the versatility and richness of mathematics, offering fresh ways to appreciate a concept that remains foundational to geometry. Here, we will delve into an elegant proof that offers a fresh perspective on this fundamental theorem. Imagine a large square with a side length of \((a+b)\), where \(a\) and \(b\) are the lengths of the two legs of a right-angled triangle. This large square, therefore, has an area of \((a+b)^2\). Within this large square, inscribe a smaller square with a side length equal to the hypotenuse \(c\) of the right-angled triangle, where \(c\) is the length of the hypotenuse. Place four right-angled triangles within the large square, each triangle having sides of length \(a\) and \(b\), and a hypotenuse of length \(c\). The arrangement of these four triangles creates a frame around the smaller square, leaving a central square in the middle with side length \(c\). The area of the large square can be calculated in two distinct ways: first, directly from its side length, and second, by summing the areas of the individual components within it. Calculating directly, the area of the large square is \((a+b)^2\). To verify this in another way, consider the arrangement of the four right-angled triangles and the small square inside the large square. Each of the four triangles has an area of \(\frac{1}{2}ab\). Hence, the total area of the four triangles is \( 4\times\frac{1}{2}ab=2ab\). The area of the small square, which has a side length of \(c\), is \(c^2\). When summing the areas of the four triangles and the central small square, the total area is \(2ab+c^2\). According to our earlier calculation, this total area should also be equal to the area of the large square, which is \((a+b)^2\). Therefore, we have the following equation:\ [(a+b)^2=2ab+c^2\] Expanding \((a+b)^2\), we get: \[ a+b)^2=a^2+2ab+b^2\] By substituting this into our earlier equation, we have \[a^2+2ab+b^2=2ab+c^2\] Subtracting \(2ab\) from both sides results in \[a^2+b^2=c^2\] This is precisely the statement of the Pythagorean Theorem, confirming that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This proof is not only visually intuitive but also highlights the intrinsic connection between the geometric arrangement and algebraic identities. By leveraging the properties of squares and right-angled triangles, we derive a proof that is both elegant and accessible, reinforcing the profound simplicity underlying one of mathematics’ most celebrated theorems. Through this approach, we gain deeper insight into the harmonious relationships between geometric shapes and algebraic expressions, showcasing the enduring beauty of mathematical proof.

Acknowledgement

None.

Conflict Of Interest

The authors are grateful to the journal editor and the anonymous reviewers for their helpful comments and suggestions.