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Transcript sketch12 Cheerful Fact
A Cheerful Fact:
The Pythagorean Theorem
Presented By: Rachel Thysell
a2 + b2 = c2
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 5th 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
a2+b2 = 9+16 = 25 =c2
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= a2+b2+2ab
Inner square has area c2, and
four triangles each with area of
½ab
Big square also equals c2+2ab
Setting them equal to each
other,
a2+b2+2ab = c2+2ab
Therefore,
a 2 + b2 = 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
AC
AH
AB
AC
and
CB
HB
AB
CB
This can be written as
AC2=ABxAH and CB2=ABxHB
Summing these two equations,
AC2+CB2=ABxAH+ABxHB=AB
x(AH+HB)=AB2
Therefore, AC2+BC2=AB2
Euclid’s Elements
Most famous proof of Pythagorean Theorem
47th 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 ka2,
kb2, and kc2
Therefore
kc2=k(a2+b2)=ka2+kb2
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
a2 + b2 = c2
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!