A Cheerful Fact: The Pythagorean Theorem

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A Cheerful Fact The Pythagorean Theorem

• Presented By Rachel Thysell

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a2 b2 c2

• Commonly known that a and b stand for the lengths
  of the shorter sides of a right triangle, and c
  is the length of the longest side, or hypotenuse

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Where did it come from?

• Often associated with Pythagoras
• Lived 5th Century B.C.
• Founder of the Pythagorean Brotherhood
• Group for learning and contemplation
• However, most commonly heard from authors who
  wrote many centuries after the time of Pythagoras

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Where did it come from?

• Found in ancient Mesopotamia, Egypt, India,
  China, and even Greece
• Known in China as Gougo Theorem
• Oldest references are from India, in the
  Sulbasutras, dating from sometime the first
  millenium B.C.
• The diagonal of a rectangle produces as much as
  is produced individually by the two sides.

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Famous Triples

• All the cultures contained triples of whole
  numbers that work as sides
• (3,4,5) is the most famous
• a2b2 916 25 c2

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It wasnt Pythagoras?

• A common discovery
• Happened during prehistoric times
• Theorem came naturally
• Independently discovered by multiple cultures
• Supported by Paulus Gerdes, cultural historian of
  mathematics
• Carefully considered patterns and decorations
  used by African artisans, and found that the
  theorem can be found in a fairly natural way

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Proofs of Pythagorean Theorem

• Whole books devoted to ways of proving the
  Pythagorean Theorem
• Many proofs found by amateur mathematicians
• U.S. President James Garfield
• He once said his mind was unusually clear and
  vigorous when studying mathematics

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Square in a Square

• Earliest proof, based on Chinese source
• Arrange four identical triangles around a square
  whose side is their hypotenuse
• Since all four triangles are identical, the inner
  quadrilateral is a square

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Square in a Square

• Big square has side ab, so area is equal to
• (ab)2 a2b22ab
• Inner square has area c2, and four triangles each
  with area of ½ab
• Big square also equals c22ab
• Setting them equal to each other,
• a2b22ab c22ab
• Therefore,
• a2 b2 c2

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Proof using Similar Triangles

• Most recent proof
• Triangles ACH and CBH are similar to ABC because
  they both have right angles and share a similar
  angle
• This can be written as AC2ABxAH and CB2ABxHB
• Summing these two equations, AC2CB2ABxAHABxHBA
  Bx(AHHB)AB2
• Therefore, AC2BC2AB2

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Euclids Elements

• Most famous proof of Pythagorean Theorem
• 47th Proposition states
• in right-angled triangles the square on the
  side opposite the right angle equals the sum of
  the squares on the sides containing the right
  triangle
• Uses areas, not lengths of the sides to prove.
• Early Greek Mathematicians did not usually use
  numbers to describe magnitudes

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Euclids Proof

• The idea is to prove that the little square (in
  blue) has the same area as the little rectangle
  (also in blue) and etc.
• He does so using basic facts about triangles,
  parallelograms, and angles.

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Euclid continues Theorem

• There is nothing special about squares in the
  theorem
• It works for any geometric figure with its base
  equal to one of the sides
• Their areas equal ka2, kb2, and kc2
• Therefore
• kc2k(a2b2)ka2kb2

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Distance Formula

• Also gave birth to the distance formula
• Makes classical coordinate geometry Euclidean
• If distance were measured some other way it would
  not be Euclidean geometry

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a2 b2 c2

• Pythagorean Theorem remains one of most important
  theorems
• One of most useful results in elementary
  geometry, both theoretically and in practice

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The End

• Any Questions?
• Thank You!