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

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

Egyptians

Egyptians

Egyptians

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

Pythagorean Triples     

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.

Proof A

A 2 +B 2 =C 2

B 2 +C 2 =A 2 A 2 =B 2 +C 2

B 2 =A 2 +C 2

Proof B We start with a right triangle.

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.

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

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

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

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.

Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

Proof B (A+B) 2 = 4 (½ (A x B)) + C 2

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

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

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

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

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

Proof B A 2 +B 2 = C 2

Proof C

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

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

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

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

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.

Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

Proof C C 2 = 4 (½ (A x B)) + (B-A) 2

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

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

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

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

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

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

Proof C C 2 = B 2 +A 2

Proof C A 2 +B 2 = C 2

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

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

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

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

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

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

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