There are many proofs to the Pythagorean Theorem. Garfield's proof, while similar to some prior proofs, is very concise and easy to understand.
Take any right-angle triangle (ABC) and make a duplicate triangle (A'AC') that is rotated and adjacent to it that you arrange as shown:
Add a line connecting points B and A'.
Now, name the sides of each triangle for the point at its opposite angle. Note, the the length A' = A and C' = C:
Garfield calculated the area of the trapezoid two ways. First, by addition of parts - adding the area of the three triangles is C2/2 + AB/2 + (AB)/2 = C2/2 + AB
The second way to calculated the area of the trapezoid is the using the formula, base times 1/2 the sum of the two sides, or: (A + B) x 1/2 x (A + B).= A2/2 + B2/2 + AB
Since both calculated areas represent the same, C2/2 + AB = A2/2 + B2/2 + AB
Subtract AB from both sides of the equation and multiply both sides by 2, the result is:
C2 = A2 + B2
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ReplyDeleteWhat is your favorite proof? In addition to the original, I like Bhaskara's first and second proofs. There are also two clever proofs called the parallelogram and broken-tiles proofs. There is a proof based on circles and the Inscribed Triangle Theorem of Thales (more to follow): https://jamesmacmath.blogspot.com/2020/04/inscribed-angle-theorem-of-thales.html
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