© T Madas It is not that we do not believe you Pythagoras but can you prove your theorem?

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Presentation transcript:

© T Madas It is not that we do not believe you Pythagoras but can you prove your theorem?

© T Madas ab Proving Pythagoras Theorem using Areas Start with a square Divide one of its sides creating lengths a and b

© T Madas a a a a b b b b Proving Pythagoras Theorem using Areas Repeat this division on all 4 sides Join the 4 division points

© T Madas a a a a b b b b Proving Pythagoras Theorem using Areas Are these four triangles congruent?

© T Madas Two triangles are congruent if… All 3 sides are equal SSS 2 sides and the contained angle are equal SAS 1 side and the 2 adjacent angles are equal ASA

© T Madas a a a a b b b b Proving Pythagoras Theorem using Areas SSS SAS ASA Are these four triangles congruent?

© T Madas a a a a b b b b Proving Pythagoras Theorem using Areas Can you prove this is a square?

© T Madas a a a a b b b b Proving Pythagoras Theorem using Areas c c c c Time for some algebra…

© T Madas Proving Pythagoras Theorem using Areas a a a a b b b b c c c c + =

© T Madas Proving Pythagoras Theorem using Areas a b c

© T Madas And I have another way of proving my theorem

© T Madas These triangles have the same base and same height… they all have the same area

© T Madas The square and the parallelogram have the same base and same height… so they have the same area 2-D shear preserves area Now watch this carefully

© T Madas

Ευκλείδης Θαλλής Πυθαγόρας The formal proof of Pythagoras theorem can be found in the most famous geometry book, Euclid’s Elements.

© T Madas A B C D θ θ 90 – θ What is the conclusion? 90 – θ

© T Madas D A B C θ θ 90 – θ 3 similar triangles

© T Madas A B C D A B C AD AB = CD AC = CB DB AB = AD AC = AB CB 3 similar triangles

© T Madas = AC CB DB AB = CB CD AC = CB DB AB = CB D A B C

© T Madas CD AC = CB DB AB = CB c AC 2 = CD x CB c AB 2 = DB x CB AC 2 +AB 2 = CD x CB + DB x CB c AC 2 +AB 2 = CB x c ( ) CD + DB AC 2 +AB 2 = CB x AC 2 +AB 2 = CB 2 D A B C

© T Madas GRAND INTERNATIONAL CHAMPION Pythagoras Miletree Owain Glyndwr x Daviot Maybe Baby. Born Lives in a wonderful home in Denmark with Annette Henricksen. … and this is how you prove my theorem properly… You’ re alwight mate?

© T Madas