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

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

3. © 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

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

5. © 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

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

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

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

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

10. © T Madas Proving Pythagoras Theorem using Areas a b c

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

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

13. © 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

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

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

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

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

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

24. © 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

25. © T Madas GRAND INTERNATIONAL CHAMPION Pythagoras Miletree Owain Glyndwr x Daviot Maybe Baby. Born 19.08.1998. Lives in a wonderful home in Denmark with Annette Henricksen. … and this is how you prove my theorem properly… You’ re alwight mate?

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