Analysis of Pythagoras Euclid’s Visual Geometry Algebraic Proofs

Analysis of Pythagoras and Euclid’s Visual Geometry and Algebraic Proofs
of the Pythagorean Theorem
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The paper presents a series of visual proofs coupled with their algebraic derivations that leads to the equation of the Pythagorean Theorem. It generally made towards those that have a weak background of mathematics that visualizes the theorem first, before jumping in into any algebraic equations. The paper makes use of Pythagoras and Euclid’s dissected proofs for the readers to further understand the concept of the Pythagorean Theorem both visually and algebraically.
According to the Pythagorean Theorem, A right triangle the square of the longest side or the hypotenuse is equal to the sum of the squares of the other two sides of the triangle containing the right angle. There are ways to solve for the Pythagorean Theorem, one of which is through the use of the algebraic formula called the quadratic formula:
x=-b±b2-4ac2a
A visual representation of the squares that forms a right triangle ABC is shown in figure 1. The right triangle has the vertex which is C and the hypotenuse or c is equal to AB. The sides that contain the right triangle are noted as a and b. In a formula representation, the square of the hypotenuse (c) is equal to the sum of the square of its sides (a2+b2) is presented as: c2 = a2+b2. However; according to Horn and Zakery (2007), many students were not aware of the multitudes of proof that established the Pythagorean Theorem. Added by Horn and Zakery (2007), many students were also not aware of the necessity of the proof as with other mathematical results, thus, the Pythagorean theorems applications and perceptions are not easily be grasped by non-mathematics students. Although algebraic proof using the formula is technical and more convincing, non-algebraic formula proofs are easier to understand and more useful to a greater range of individuals regardless of mathematical background. This paper aims to use a series of algebraic proofs together with the use of explained geometric proofs that requires less algebraic knowledge to be understood by students with problematic mathematical backgrounds.
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Figure 1: Diagrammatic notations of a right triangle showing the squares of each side of the Pythagorean Theorem
The first proof was used by Pythagoras (fig. 2) is a visual proof that doesn’t need any algebraic structures but was used to support the evidences of the Pythagorean Theorem. In the figure, it is seen that a set of right triangles were arranged to form different p...