Pythagorean Theorem Inequality and Pythagorean Triples
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Pythagorean Theorem Inequality
Used to classify triangles by angles
Longest side < short side + short side - ACUTE triangle
Longest side = short side + short side - RIGHT triangle
Longest side > short side + short side - OBTUSE triangle
The Pythagorean Theorem describes the relationship between the sides of a right triangle.leg + leg = hypotenuseorshort side + short side = long sideA Pythagorean triple is a set of integers, a, b, and c, that could be the sides of a right triangle if a + b = c.
7, 24, 25If not, then what kind of triangle is it? Or, is it not a triangle at all?8, 9, 1060, 11, 6133, 42, 981, 1, 8273, 19, 18
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.
nn - 1 2n + 1 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
nn - 1 2n + 1 2,,Find the sides of a Pythagoras triangle if n = 3.3, 4, 5Find the sides of a Pythagoras triangle if n = 2.2, 3/2, 5/2Why might you want to restrict n to odd positive integers in Pythagorass formula?Pythagoras, contorniate medallion engraved between AD 395 and 410
aa4,,- 1a4+ 1It 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.
aa4,,- 1a4+ 1Find the sides of a Plato triangle if a = 4.3, 4, 5Find the sides of a Plato triangle if a = 7.11.25, 7, 13.25Why might you want to restrict values of a to even positive integers greater than 2 in Platos formula?
xyx - y 2x + y 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.
Find the sides of a Euclid triangle if x = 3 and y = 1.1, 3 , 2Find the sides of a Euclid triangle if x = 10 and y = 4.3, 40 , 7Why might you want to restrict values of x and y to either even or odd numbers in Euclids formula?xyx - y 2x + y 2,,Find the sides of a Euclid triangle if x = 5 and y = 2.3/2, 10 , 7/2
2pqp - q,,p + qMaseres 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.
Find the sides of a Masres triangle if p = 4 and q = 1.8, 15, 17Find the sides of a Masres triangle if p = 2.6 and q = 188.8.131.52 , 4.51, 9.01What restriction would you impose on values for p and q in Masres formula?2pqp - q,,p + q2pqp - qp + q
Finding Pythagorean TriplesPythagorean Triple - A set of three whole numbers such that a + b = cPythagoras formulaPlatos formula
-use odd positive integers-even positive integers greaterthan 2
Euclids formulaMaseres formulann - 1 2n + 1 2,,aa4,,- 1a4+ 1xyx - y 2x + y 2,,2pqp - q,,p + qCOLORED NOTE CARD-both even or both odd, not always a triple-Whole numbers
. . . one number equal to 16.
. . . one number equal to 17.
. . . the numbers 9 and 7.
. . . the numbers 5 and 6.