Proof of the Pythagorean TheoremPresident Garfield found a proof of the Pythagorean Theorem. Here is how it goes! Start with two right triangles with legs a and b, and hypotenuse c. Notice that it is the same triangle You just have to rotate the triangle on the right 90 degrees counterclockwise. Then move it to the left until it touches the triangle on the left Therefore, the area of the 3 triangles must equal the area of the trapezoid The area of triangle ABC is (base × height ) / 2 The area of triangle ABC = (a × b ) / 2 The triangle on the right is the same triangle, so the area is also (a × b ) / 2 Now, what about the triangle in the middle or the one with a red side? Notice that angle m + angle n = 90 degrees. And m + n + f = 180 degrees 90 + f = 180 f = 90 degrees The triangle in middle is then a right triangle. Thus, the height is also equal to the base that is c and Area = ( c × c ) / 2 = c^{2} / 2 Area of trapezoid = h/2 × (b_{1} + b_{2}) Area of trapezoid = ((a + b)/2) × (a + b) Finally, translate "the area of the 3 triangles must equal the area of the trapezoid" into an equation (a × b ) / 2 + (a × b ) / 2 + c^{2} / 2 = ((a + b) / 2) × (a + b) (a × b + a × b ) / 2 + c^{2} / 2 = ((a + b) / 2) × (a + b) (2 a × b) / 2 + c^{2} / 2 = (a + b)^{2} / 2 Multiply everything by 2 2 a × b + c^{2} = (a + b)^{2} 2 a × b + c^{2} = a^{2} + 2 a × b + b^{2} Subtract 2 a × b from both sides: c^{2} = a^{2} + b^{2} Proof of the pythagorean Theorem is complete! 




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