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4 5 Pythagoras theorem a proof by dissection The Theorem of Pythagoras ca. 569 475 B.C.E. states that in a right angled triangle the square on the hypotenuse, or long side, is equal to the sum of the squares on the other two sides opposite, top. Nowadays this is written algebraically as a 2 b 2 c 2. Proof Arrange four identical rightangled triangles with sides a, b and c in a large square of side a b, leaving two square spaces with sides a and b, respectively opposite, middle left. The four triangles can also be arranged in this large square to leave a central square space with side c opposite, middle right. In both cases the contained squares equal the large square minus four times the triangle. Therefore the sum of the two smaller squares, a 2 b 2, equals the larger square, c 2. Q.E.D. Conversely, and this requires an extra proof, IF a triangles sides are related as above, THEN it is rightangled. Integers which satisfy the equality a 2 b 2 c 2 are known as Pythagorean triples. An ancient construction of a right angle from a loop of string with 3 4 5 12 equally spaced knots is based on the triple 345 below left. A Babylonian clay tablet, Plimpton 322, lists integer pairs corresponding to Pythagorean triples below right, which suggests that our general theorem may well have been known long before Pythagoras.
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