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Exploring Pythagorean Theorem Variations for Next Year’s Math Contest T-shirt

Dive into the rich history and geometric proofs related to Pythagorean Theorem variations, including modernized proofs by Euclid and insights into Pythagorean triples. Discover the fascinating world of mathematical patterns!

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Exploring Pythagorean Theorem Variations for Next Year’s Math Contest T-shirt

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  1. What Should AdornNext Year’s MathContest T-shirt? By Kevin Ferland

  2. b a b c a c c a b c a b Last Year’s T-shirt total area = area of inner square + area of 4 triangles

  3. Proof Pythagoras 585-500 B.C.E. The t-shirt proof is believed to be the type used by the Pythagoreans.

  4. c b a Pythagorean Theorem Given a right triangle we have

  5. b a c c b a James Garfield 1876 total area = area of inner half-square + area of 2 triangles

  6. I II a b x c-x I II ← c → Euclid 323-285 B.C.E.(modernized) proof from Elements

  7. The result was known by Babylonian mathematicians circa 1800 B.C.E.

  8. The oldest known proof is found in a Chinese text circa 600 B.C.E.

  9. January 2010

  10. b a b c a c c a b c a b Last Year’s T-shirt total area = area of inner square + area of 4 triangles

  11. b a 120° 120° b c a c a b c 120° 120° c b a c a c b 120° 120° a b Generalizing the t-shirt total area = area of inner hexagon + area of 6 triangles

  12. FACT: The area of a regular hexagon with side length s is • FACT: The area of a 120°-triangle with (short) sides a and b is

  13. Proof

  14. c b 120° a Result Given a 120°-triangle we have

  15. b a b c a c a b c

  16. STOP?

  17. DON’T STOP Clearly, this argument extends to any regular 2k-gon for k ≥ 2.

  18. c b θ a Result (n = 2k) Given a θ-triangle we have

  19. s θ/2 θ/2 s s θ s s θ s s θ s FACT: The area of a regular n-gon with side length s is

  20. s/2 s/2 θ/2 h

  21. c h b θ a FACT: The area of a θ-triangle with (short) sidesa and bis

  22. Trig Identity: Proof

  23. b a b c a c a b c Generalized Pythagorean Theorem total n-gon area = area of inner n-gon + area of n θ-triangles

  24. Proof

  25. c b θ a Result Given a θ-triangle we have TheLAW OF COSINESin these cases.

  26. Pythagorean Triples A triple (a, b, c) of positive integers such that a2 + b2 = c2 is called a Pythagorean triple. It is called primitive if a, b, and c are relatively prime.

  27. 5 3 4 E.g. Whereas, (6, 8, 10) is a Pythagorean triple that is not primitive.

  28. Theorem: All primitive solutions to a2 + b2 = c2 (satisfying a even and b consequently odd) are given by where

  29. 7 3 120° 5 E.g. n = 6, θ = 120° What is the characterization of all such primitive triples?

  30. Other 120°-triples (7, 8, 13), (5, 16, 19), … There does exist a characterization of these. What can you find?

  31. b a 120° 120° b c a c a b c 120° 120° c b a c a c b 120° 120° a b Next Year’s t-shirt

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