Pythagorean triples
- 1. P YTHAGOREAN T RIPLES
- 2. P YTHAGOREAN HISTORY The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (569-500 B.C.), who was perhaps the first to offer a proof of the theorem. The Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypotenuse, or in mathematical terms for the triangle shown at the bottom, a2 + b2 = c2. Integers that satisfy the conditions a2 + b2 = c2 are called ‘Pythagorean triples’
- 3. P YTHAGOREAN H ISTORY: C ONTINUED Ancient Indian mathematicians also knew the Pythagorean theorem, and the Sulbasutras (of which the earliest date from ca. 800-600 B.C.) discuss it in the context of strict requirements for the orientation, shape, and area of altars for religious purposes. It has also been suggested that the ancient Mayas used variations of Pythagorean triples in their Long Count calendar.
- 5. E XAMPLE : PART 1 The simplest triangle with sides of integer length is the 3-4-5 triangle. The numbers 3,4 and 5 satisfy the rule 3²+4²=5² This is worked out like this: 3²=9, 4²=16 and 5²=25 so 9+16=25 The set of numbers (3,4,5) is called a Pythagorean Triple because since it obeys the a²+b²=c² rule.
- 6. E XAMPLE : PART 2 There are other triangles which do not obey the Pythagorean Triple rule, for instance the (3-7-9) triangle does not obey the rule because: 3²=9, 7²=49 and 9²=81, but 9+49=58, there is a difference of 23 The set of numbers (3-7-9) is not classified as a Pythagorean triple. There are many triangles just like the one shown, they do not satisfy the Pythagorean Triple Rule.
- 8. Q UESTIONS 1) Show that {3, 4, 5} is a Pythagorean Triple? 2) Is {6, 8, 10} a Pythagorean Triple? Explain why? 3) Is {9, 12, 15} a Pythagorean Triple? Explain why? 4) Find x if {18, 24, x} is a Pythagorean Triple?
- 9. U SES IN B UILDINGS The picture to the bottom shows excavating putting in flooring deals with squares and triangles using the Pythagorean Theorem. Some builders use this formula, because they can find the missing sides. So as you can see, the Pythagorean Theorem is used in building houses and buildings.
- 11. U SES IN E VERYDAY L IFE Find the answers about the baseball diamond e.g. How far the catcher has to throw the ball to get from home plate to second base. What type of ladder you will need to get to your roof You could use it to determine the distance between two places.
- 12. U SES IN A RT Pythagoras greatly admired the arts and even thought of himself as an artist. Artists use the triangle as pieces in a mosaic or the shape in drawings. Some artists use the triangle as a drawing tool. Pythagoras had created the triangle and the ideas on how to use it.