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Proof of the Pythagorean Theorem


President 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


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Put the two triangles next to each other so that legs a and b form a straight line.

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


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Connect the endpoints to make a trapezoid like the one you see below:


Proof-of-the-pythagorean-Theorem-image


Label the angles inside the triangles as shown below. This will help us prove that the triangle in the middle (one side is red) is a right triangle


Proof-of-the-pythagorean-Theorem-image


The important thing here to notice is that there are 3 triangles and put together, these triangles form a trapezoid

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 = c2 / 2

Area of trapezoid = h/2 × (b1 + b2)

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 + c2 / 2 = ((a + b) / 2) × (a + b)

(a × b + a × b ) / 2 + c2 / 2 = ((a + b) / 2) × (a + b)

(2 a × b) / 2 + c2 / 2 = (a + b)2 / 2

Multiply everything by 2

2 a × b + c2 = (a + b)2

2 a × b + c2 = a2 + 2 a × b + b2

Subtract 2 a × b from both sides:

c2 = a2 + b2

Proof of the pythagorean Theorem is complete!






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