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The Pythagorean Theorem, also known as Euclid 1.47 (i.e., Proposition 47 in Book / of the Elements), says that the areas of the squares built on the catheti of a right triangle add up to the area of the square built on the hypotenuse: A + B ?
A “Cut and Paste” Geometric Proof of the Pythagorean Theorem. Draw a right triangle (shown in orange above) with squares on its sides, shown in white, green and blue above. We are going to show that the green and blue squares on the triangle’s two legs can be cut up and fit into the white square on the triangle’s hypotenuse.
The Pythagorean theorem is a fundamental result in Euclidean geometry that relates the side lengths of a right triangle through the simple relationship a² + b² = c². The relationship was known to the ancient scholars and builders of Babylon, Egypt, China and India.
In the figure, the triangles whose are areas are marked x and y are similar to the original triangle (which has area x+y). So accepting that areas of similar right-angled triangles are proportional to the squares of the hypotenuse, x:y:x+y are in ratio a 2:b 2:c 2, which is Pythagoras's theorem.
2 ημέρες πριν · Pythagorean theorem, geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse. Although the theorem has long been associated with the Greek mathematician Pythagoras, it is actually far older.
Hundreds of thousands of clay tablets, found over the past two centuries, attest to a people who flourished in commerce and architecture, kept accurate records of astronomical events, excelled in the arts and literature, and, under the rule of Hammurabi, created the first legal code in history.
The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate.
