1 / 33

What Should Adorn Next Year’s Math Contest T-shirt?

What Should Adorn Next Year’s Math Contest T-shirt?. By Kevin Ferland. 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. Proof. Pythagoras 585-500 B.C.E.

metea
Download Presentation

What Should Adorn Next Year’s Math Contest T-shirt?

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related