The Pythagorean The Pythagorean TheoremTheorem

Everything you need to know Everything you need to know about the sides of a right about the sides of a right

triangletriangle

HistoryHistory

Pythagoras (569-500 B.C.E.) was born on Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece.the island of Samos in Greece.

Pythagoras founded the Brotherhood of Pythagoras founded the Brotherhood of Pythagoreans, devoted to the study of math.Pythagoreans, devoted to the study of math.

believed that "Number rules the believed that "Number rules the universe,"and gave numerical values to universe,"and gave numerical values to many objects and ideas.many objects and ideas.

Legend has it that upon completion of his Legend has it that upon completion of his famous theorem, Pythagoras sacrificed 100 famous theorem, Pythagoras sacrificed 100 oxen oxen

Cont.Cont. Not possible to tell if Pythagoras is Not possible to tell if Pythagoras is

the actual author of the Pythagorean the actual author of the Pythagorean TheoremTheorem

Known since the days of the ancient Known since the days of the ancient Babylonians and EgyptiansBabylonians and Egyptians

4000 year old Babylonian tablet (c. 4000 year old Babylonian tablet (c. 1900 B.C.E.), known as 1900 B.C.E.), known as PlimptonPlimpton 322 322, lists columns of numbers , lists columns of numbers showing what we now call showing what we now call Pythagorean TriplesPythagorean Triples

Plimpton 322Plimpton 322

Pythagorean Triples are sets of Pythagorean Triples are sets of numbers that satisfy the numbers that satisfy the

equationequation

aa22+b+b22==cc22

What does it mean?What does it mean?

Certain numbers satisfy this equation, Certain numbers satisfy this equation, like a=3, b=4, and c=5.like a=3, b=4, and c=5.

These numbers may not seem that special These numbers may not seem that special at first, but is there a source that they all at first, but is there a source that they all come from?come from?

The answers is YES, they come from right The answers is YES, they come from right triangles, and a, b, and c represent the triangles, and a, b, and c represent the legslegs

Lets see how the relationship is Lets see how the relationship is discovereddiscovered

The PictureThe Picture

The GoalThe Goal

Find two different ways to express the Find two different ways to express the area of the square from the previous slidearea of the square from the previous slide

Once we have two ways to express the Once we have two ways to express the area, set them equal to each otherarea, set them equal to each other

Simplify the equation as much as possible Simplify the equation as much as possible andand

SEE WHAT HAPPENSSEE WHAT HAPPENS

First WayFirst Way

a+b

a+b

Area = (a+b)2

Second WaySecond Way

ba21

• 4 triangles with area

•The center square with area = c2

So the area of the square is also

2

221

2

4

cba

cba

Put them togetherPut them together Set both areas equal to each other and simplifySet both areas equal to each other and simplify

222

222

22

22

2

cba

cabbaba

cabba

Bring it togetherBring it together

The numbers are special and they do The numbers are special and they do come from something geometricalcome from something geometrical

The Pythagorean Theorem has been The Pythagorean Theorem has been known for centuriesknown for centuries

It is still a very useful formula todayIt is still a very useful formula today

Problems to work onProblems to work on

Suppose a=4, b=6, Suppose a=4, b=6, what is c?what is c?

Suppose a=3, c=8, Suppose a=3, c=8, what is b?what is b?

Suppose b=5, Suppose b=5, c=13, what is a?c=13, what is a?