A Cheerful Fact: The Pythagorean Theorem Presented By: Rachel Thysell

a 2 + b 2 = c 2 Commonly known that a and b stand for the lengths of the shorter sides of a right triangle, and c is the length of the longest side, or hypotenuse

Where did it come from? Often associated with Pythagoras  Lived 5 th Century B.C.  Founder of the Pythagorean Brotherhood Group for learning and contemplation However, most commonly heard from authors who wrote many centuries after the time of Pythagoras

Where did it come from? Found in ancient Mesopotamia, Egypt, India, China, and even Greece  Known in China as “Gougo Theorem”  Oldest references are from India, in the Sulbasutras, dating from sometime the first millenium B.C.  The diagonal of a rectangle “produces as much as is produced individually by the two sides.”

Famous Triples All the cultures contained “triples” of whole numbers that work as sides (3,4,5) is the most famous a 2 +b 2 = 9+16 = 25 =c 2

It wasn’t Pythagoras? A common discovery  Happened during prehistoric times Theorem came “naturally”  Independently discovered by multiple cultures  Supported by Paulus Gerdes, cultural historian of mathematics Carefully considered patterns and decorations used by African artisans, and found that the theorem can be found in a fairly natural way

Proofs of Pythagorean Theorem Whole books devoted to ways of proving the Pythagorean Theorem Many proofs found by amateur mathematicians  U.S. President James Garfield  He once said his mind was “unusually clear and vigorous” when studying mathematics

“Square in a Square” Earliest proof, based on Chinese source Arrange four identical triangles around a square whose side is their hypotenuse Since all four triangles are identical, the inner quadrilateral is a square

“Square in a Square” Big square has side a+b, so area is equal to (a+b) 2 = a 2 +b 2 +2ab Inner square has area c 2, and four triangles each with area of ½ab Big square also equals c 2 +2ab Setting them equal to each other, a 2 +b 2 +2ab = c 2 +2ab Therefore, a 2 + b 2 = c 2

Proof using Similar Triangles Most recent proof Triangles ACH and CBH are similar to ABC because they both have right angles and share a similar angle This can be written as AC 2 =ABxAH and CB 2 =ABxHB Summing these two equations, AC 2 +CB 2 =ABxAH+ABxHB=AB x(AH+HB)=AB 2 Therefore, AC 2 +BC 2 =AB 2

Euclid’s Elements Most famous proof of Pythagorean Theorem 47 th Proposition states: “in right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right triangle” Uses areas, not lengths of the sides to prove.  Early Greek Mathematicians did not usually use numbers to describe magnitudes

Euclid’s Proof The idea is to prove that the little square (in blue) has the same area as the little rectangle (also in blue) and etc. He does so using basic facts about triangles, parallelograms, and angles.

Euclid continues Theorem There is nothing special about “squares” in the theorem It works for any geometric figure with its base equal to one of the sides Their areas equal ka 2, kb 2, and kc 2 Therefore kc 2 =k(a 2 +b 2 )=ka 2 +kb 2

Distance Formula Also gave birth to the distance formula Makes classical coordinate geometry “Euclidean”  If distance were measured some other way it would not be Euclidean geometry

a 2 + b 2 = c 2 Pythagorean Theorem remains one of most important theorems One of most useful results in elementary geometry, both theoretically and in practice

The End Any Questions? Thank You!