1 / 15

Pythagorean Theorem Inequality and Pythagorean Triples

Pythagorean Theorem Inequality and Pythagorean Triples. Pythagorean Theorem Inequality Used to classify triangles by angles Longest side ² < short side ² + short side² - ACUTE triangle Longest side² = short side² + short side² - RIGHT triangle

nira
Télécharger la présentation

Pythagorean Theorem Inequality and Pythagorean Triples

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Pythagorean Theorem Inequality and Pythagorean Triples

  2. 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

  3. Pythagorean Theorem 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.

  4. Pythagorean Triples or Not? 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

  5. 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.

  6. Pythagoras' Formula n² - 1 2 n² + 1 2 n , , 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

  7. Pythagoras' Formula n² - 1 2 n² + 1 2 n , , Find the sides of a Pythagoras triangle if n = 3. 3, 4, 5 Pythagoras, contorniate medallion engraved between AD 395 and 410 Find the sides of a Pythagoras triangle if n = 2. 2, 3/2, 5/2 Why might you want to restrict n to odd positive integers in Pythagoras’s formula?

  8. Plato's Formula a² 4 a² 4 - 1 , a , + 1 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.

  9. Plato's Formula a² 4 a² 4 - 1 , a , + 1 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?

  10. Euclid's Formula x - y 2 x + y 2 , xy , 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.

  11. Euclid's Formula x - y 2 x + y 2 , xy , 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?

  12. Maseres' 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.

  13. Maseres' Formula 2pq , p² - q² , p² + q² p² + q² 2pq Find the sides of a Masères triangle if p = 4 and q = 1. 8, 15, 17 p² - q² Find the sides of a Masères triangle if p = 2.6 and q = 1.5. 7.8 , 4.51, 9.01 What restriction would you impose on values for p and q in Masères’ formula?

  14. COLORED NOTE CARD Finding Pythagorean Triples Pythagorean Triple - A set of three whole numbers such that a² + b² = c² Pythagoras’ formula Plato’s formula -use odd positive integers -even positive integers greater than 2 Euclid’s formula Maseres’ formula n² + 1 2 a² 4 a² 4 n² - 1 2 n - 1 , a , + 1 , , x - y 2 x + y 2 2pq , p² - q² , p² + q² , xy , -Whole numbers -both even or both odd, not always a triple

  15. Find a Pythagorean Triple with . . . . . . one number equal to 16. . . . one number equal to 17. . . . the numbers 9 and 7. . . . the numbers 5 and 6.

More Related