2. Pythagorean Theorem A 2 +B 2 =C 2 Michelle Moard

3. Egyptians  How were the pyramid’s built? (…and so precise?)

4. Egyptians

5. Egyptians

6. Egyptians

7. Pythagorean Cult  Lead by Pythagoras of Samos (570-490 B.C.)  Believed that everything in nature is related to math and can be predicted  Swore to secrecy and strict loyalty

8. Pythagorean Triples  3-4-5  5-12-13  7-24-25  9-40-41  11-60-61

9. Proofs  There are numerous proofs of the Pythagorean Theorem from algebra and geometry and beyond.  Today, I will go through three of my favorite proofs.

10. Proof A

26. A 2 +B 2 =C 2

27. B 2 +C 2 =A 2 A 2 =B 2 +C 2

28. B 2 =A 2 +C 2

29. Proof B We start with a right triangle.

30. Proof B We construct a square by placing four congruent triangles in a manner such that the hypotenuse creates its own smaller square in the center of the larger square.

31. Proof B We can see that the area of the large square is (A+B) x (A+B) or simply (A+B) 2

32. Proof B The area of the small square is C x C or C 2

33. Proof B The area of the original triangle is ½ (A x B)

34. Proof B We can see that the area of the large square is the sum of four triangles and the area of the small, square.

35. Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

36. Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

37. Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

38. Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

39. Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

40. Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

41. Proof B (A+B) 2 = 2AB + C 2

42. Proof B (A+B) 2 = 2AB + C 2

43. Proof B A 2 +2AB+B 2 = 2AB + C 2

44. Proof B A 2 +2AB+B 2 = 2AB + C 2

45. Proof B A 2 +2AB+B 2 = 2AB + C 2

46. Proof B A 2 +B 2 = C 2

48. Proof C

49. We construct a square by placing four congruent triangles in a manner such that the hypotenuse is the perimeter of the large square.

50. Proof C The area of the large square is C x C or C 2.

51. Proof C The area of the small square is (B-A) x (B-A) or (B-A) 2.

52. Proof C The area of the original triangle is ½ (A x B).

53. Proof C We can see that the area of the large square is the sum of the four small triangles and the small square in the center.

54. Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

55. Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

56. Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

57. Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

58. Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

59. Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

60. Proof C C 2 = 2AB + (B-A) 2

61. Proof C C 2 = 2AB + (B-A) 2

62. Proof C C 2 = 2AB+ (B-A) 2

63. Proof C C 2 = 2AB + B 2 -2AB +A 2

64. Proof C C 2 = 2AB + B 2 -2AB +A 2

65. Proof C C 2 = 2AB + B 2 -2AB +A 2

66. Proof C C 2 = B 2 +A 2

67. Proof C A 2 +B 2 = C 2

69. Why is the Pythagorean Theorem so Important?  Constructing 90 degree angles  Right angles are used everywhere from building construction to trigonometric functions

70. Does the Pythagorean Theorem apply to other powers?  What about A 3 +B 3 =C 3?  What about A 4 +B 4 =C 4 ?

71. Does the Pythagorean Theorem apply to other powers?  What about A 3 +B 3 =C 3?  What about A 4 +B 4 =C 4?  For what values of x can we find an a, b and c so that the following statement is true? A x +B x =C x

72. Does the Pythagorean Theorem apply to other powers? A x +B x =C x ? X=?

73. Does the Pythagorean Theorem apply to other powers? Andrew Wiles proved in 1993, that A x +B x =C x only works when X=2

74. Pythagorean Theorem A 2 +B 2 =C 2 Michelle Moard

75. Resources Used & other good sites  Math 128 Modern Geometry Link Link  Dr. Peggie House  Pythagorean Theorem Link Link  Pythagorean Theorem Applet Link Link  Pythagoras Link Link  Wikipedia Link Link