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Malhar Rajpal Most of us believe this fascinating finding without even questioning whether it is really true for ALL right angled triangles, and if it is, why it actually works. In this article, I will be showing you my favorite proof of the Pythagorean theorem! Let’s say we have a square with side length a+b. We can split each side into length a and length b as shown in the diagram above. If we connect each of the splitting points (the dot in between a and b on both sides), we can see 4 right triangles which are similar since they all have sides a and b and have a right angle, as well as an additional quadrilateral area. Because the triangles are similar , we can say the third side of each of these triangles is the same length, which we can call c. This is shown in the diagram below: To show that the quadrilateral in the centre of the diagram is a square, we need to show that it has equal side lengths (which we have already done, each side has length c), but we also need to show that it has equal angles. Since all four ABC triangles are similar and right angled, we can say that each angle enclosed by sides b and c is called θ and thus the side enclosed by the sides a and c must be equal to 180 - 90 - θ = 90 - θ, since the angles of a triangle must add up to 180 degrees. This is shown in the next diagram: Since each side on the external square is a straight line, we can say that 90 - θ + θ + x = 180, with x being one angle of the quadrilateral surrounded by the lengths c. By solving this, we can see that x = 90 degrees, a right angle. Since there are four sides and four angles, and since all of them must be x, 90 degrees, the quadrilateral is thus proven to be a square: So now we can compare areas. We can see the entire area of the external square is (a+b)^2 since each side of the external square is a+b. We can also represent this area as the sum of the areas of the four triangles with side lengths a, b and c and the square with side length c.
The area of the triangles is 4*(1/2)*a*b = 2ab The area of the square is c^2 So we can say (a+b)^2 = c^2 + 2ab Expanding the LHS gives a^2 + 2ab + b^2 = c^2 + 2ab, which cancels to a^2 + b^2 = c^2. This is Pythagoras' theorem! Although the theorem is named the Pythagorean Theorem after the famous Greek philosopher Pythagoras, its actual roots are actually unknown and widely debated. I have shown you one proof of the Pythagorean theorem in this article, but there are actually numerous different proofs, ranging from complex geometric proofs by extremely famed mathematicians to Einstein’s proof by dissection without rearrangement, and proofs using algebra and differentials! The uses for Pythagorean theorem are innumerable and that is why it is so famous! After all, if you are going to use it throughout your secondary education and beyond, you should know why it works!
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