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Pythagorean Theorem Inequality Used to classify triangles by angles Longest side² < short side² + short side² - ACUTE triangle Longest side² = short side² + short side² - RIGHT triangle Longest side² > short side² + short side² - OBTUSE triangle The Pythagorean Theorem describes the relationship between the sides of a right triangle. leg² + leg² = hypotenuse² or short side² + short side² = long side² A Pythagorean triple is a set of integers, a, b, and c, that could be the sides of a right triangle if a² + b² = c². If not, then what kind of triangle is it? Or, is it not a triangle at all? 7, 24, 25 8, 9, 10 60, 11, 61 73, 19, 18 81, 1, 82 33, 42, 9 Many mathematicians over the centuries have developed formulas for generating side lengths for right triangles. Some generate Pythagorean triples, others just generate the side lengths for a right triangle. Masères n , n² - 1 , n² + 1 2 2 Of course today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years earlier he may have been the first to prove it. Number rules the universe. -Pythagoras n , n² - 1 , n² + 1 2 2 Find the sides of a Pythagoras triangle if n = 3. 3, 4, 5 Find the sides of a Pythagoras triangle if n = 2. 2, 3/2, 5/2 Pythagoras, contorniate medallion engraved between AD 395 and 410 Why might you want to restrict n to odd positive integers in Pythagoras’s formula? a² - 1 , a , a² + 1 4 4 It was claimed that Plato's real name was Aristocles, and that 'Plato' was a nickname (roughly 'the broad') derived either from the width of his shoulders, the results of training for wrestling, or from the size of his forehead. Although Plato made no important mathematical discoveries himself, his belief that mathematics provides the finest training for the mind was extremely important in the development of the subject. Over the door of the Academy was written:Let no one unversed in geometry enter here. a² - 1 , a , a² + 1 4 4 Find the sides of a Plato triangle if a = 4. 3, 4, 5 Find the sides of a Plato triangle if a = 7. 11.25, 7, 13.25 Why might you want to restrict values of a to even positive integers greater than 2 in Plato’s formula? x - y , xy , x + y 2 2 Euclid's most famous work is his treatise on mathematics The Elements. The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly due to him. x - y , xy , x + y 2 2 Find the sides of a Euclid triangle if x = 3 and y = 1. 1, 3 , 2 Find the sides of a Euclid triangle if x = 10 and y = 4. 3, 40 , 7 Find the sides of a Euclid triangle if x = 5 and y = 2. 3/2, 10 , 7/2 Why might you want to restrict values of x and y to either even or odd numbers in Euclid’s formula? 2pq , p² - q² , p² + q² Maseres wrote many mathematical works which show a complete lack of creative ability. He rejected negative numbers and that part of algebra which is not arithmetic. It is probable that Maseres rejected all mathematics which he could not understand. Masères 2pq , p² - q² , p² + q² Find the sides of a Masères triangle 2pq if p = 4 and q = 1. 8, 15, 17 Find the sides of a Masères triangle if p = 2.6 and q = 1.5. 7.8 , 4.51, 9.01 p² - q² What restriction would you impose on values for p and q in Masères’ formula? COLORED NOTE CARD Finding Pythagorean Triples Pythagorean Triple - A set of three whole numbers such that a² + b² = c² Pythagoras’ formula Plato’s formula n² + 1 a² a² n² - 1 n , -1 , a , +1 , 2 4 4 2 -use odd positive integers -even positive integers greater than 2 Euclid’s formula Maseres’ formula x - y , xy 2 , x + y 2 -both even or both odd, not always a triple 2pq , p² - q² , p² + q² -Whole numbers . . . one number equal to 16. . . . one number equal to 17. . . . the numbers 9 and 7. . . . the numbers 5 and 6.