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Pythagorean Pythagorean TheoremTheorem
AA22+B+B22=C=C22
Michelle MoardMichelle Moard
Egyptians Egyptians
How were the pyramid’s built?How were the pyramid’s built?
(…and so precise?)(…and so precise?)
Egyptians Egyptians
Egyptians Egyptians
Egyptians Egyptians
90°
Pythagorean CultPythagorean Cult
Lead by Pythagoras of Samos Lead by Pythagoras of Samos (570-490 B.C.)(570-490 B.C.)
Believed thatBelieved that everything in nature is everything in nature is related torelated to math and can be predictedmath and can be predicted
Swore to secrecy and strict loyaltySwore to secrecy and strict loyalty
Pythagorean TriplesPythagorean Triples
3-4-53-4-5 5-12-135-12-13 7-24-257-24-25 9-40-419-40-41 11-60-6111-60-61
ProofsProofs
There are numerous proofs of the There are numerous proofs of the Pythagorean Theorem from algebra and Pythagorean Theorem from algebra and geometry and beyond. geometry and beyond.
Today, I will go through three of my Today, I will go through three of my favorite proofs. favorite proofs.
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
Proof AProof A
A
B
C
A2+B2=C2
C
AB
B2+C2=A2
A2=B2+C2
C
B
A
B2=A2+C2
C
B
A
Proof BProof B
A
B
C
We start with a right triangle.
Proof BProof B
A
B
C
A
B
C
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 BProof B
A
B
C
A
B
C
We can see that the area of the large square is (A+B) x (A+B)
or simply (A+B)2
Proof BProof B
A
B
C
A
B
C
The area of the small square is C x C
or C2
Proof BProof B
A
B
C
A
B
C
The area of the original triangle is
½ (A x B)
Proof BProof B
A
B
C
A
B
C
We can see that the area of the large square is the sum of four triangles and the area of the small, square.
Proof BProof B
A
B
C
A
B
C
(A+B)2 = 4 (½ (A x B)) + C2
Proof BProof B
A
B
C
A
B
C
(A+B)2 = 4 (½ (A x B)) + C2
Proof BProof B
A
B
C
A
B
C
(A+B)2 = 4 (½ (A x B)) + C2
Proof BProof B
A
B
C
A
B
C
(A+B)2 = 4 (½ (A x B)) + C2
Proof BProof B
A
B
C
A
B
C
(A+B)2 = 4 (½ (A x B)) + C2
Proof BProof B
A
B
C
A
B
C
(A+B)2 = 4 (½ (A x B)) + C2
Proof BProof B
A
B
C
A
B
C
(A+B)2 = 2AB + C2
Proof BProof B
A
B
C
A
B
C
(A+B)2 = 2AB + C2
Proof BProof B
A
B
C
A
B
C
A2+2AB+B2 = 2AB + C2
Proof BProof B
A
B
C
A
B
C
A2+2AB+B2 = 2AB + C2
Proof BProof B
A
B
C
A
B
C
A2+2AB+B2 = 2AB + C2
Proof BProof B
A
B
C
A
B
C
A2+B2 = C2
Proof CProof C
AC
B
Proof CProof C
AC
C
B-A
B
We construct a square by placing four congruent triangles in a manner such that the hypotenuse is
the perimeter of the large square.
Proof CProof C
AC
C
B-A
B
The area of the large square is C x C
or C2.
Proof CProof C
AC
C
B-A
B
The area of the small square is (B-A) x (B-A)
or (B-A)2.
Proof CProof C
AC
C
B-A
B
The area of the original triangle is ½ (A x B).
Proof CProof C
AC
C
B-A
B
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 CProof C
AC
C
B-A
B
C2= 4 (½ (A x B)) + (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 4 (½ (A x B)) + (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 4 (½ (A x B)) + (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 4 (½ (A x B)) + (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 4 (½ (A x B)) + (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 4 (½ (A x B)) + (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 2AB + (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 2AB + (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 2AB+ (B-A)2
Proof CProof C
AC
C
B-A
B
C2= 2AB + B2-2AB +A2
Proof CProof C
AC
C
B-A
B
C2= 2AB + B2-2AB +A2
Proof CProof C
AC
C
B-A
B
C2= 2AB + B2-2AB +A2
Proof CProof C
AC
C
B-A
B
C2= B2+A2
Proof CProof C
AC
C
B-A
B
A2 +B2= C2
Why is the Pythagorean Why is the Pythagorean Theorem so Important?Theorem so Important?
Constructing 90 degree anglesConstructing 90 degree angles Right angles are used everywhere from Right angles are used everywhere from
building construction to trigonometric building construction to trigonometric functionsfunctions
Does the Pythagorean Does the Pythagorean Theorem apply to other Theorem apply to other powers?powers?
What about AWhat about A33+B+B33=C=C3?3?
What about AWhat about A44+B+B44=C=C44??
Does the Pythagorean Does the Pythagorean Theorem apply to other Theorem apply to other powers?powers?
What about AWhat about A33+B+B33=C=C3?3?
What about What about AA44+B+B44=C=C4?4?
For what values of x can we find an a, b For what values of x can we find an a, b and c so that the following statement is and c so that the following statement is true?true?
AAxx+B+Bxx=C=Cxx
Does the Pythagorean Does the Pythagorean Theorem apply to other Theorem apply to other powers?powers?
AAxx+B+Bxx=C=Cxx??
X=?X=?
Does the Pythagorean Does the Pythagorean Theorem apply to other Theorem apply to other powers?powers?
Andrew Wiles proved in 1993, that Andrew Wiles proved in 1993, that AAxx+B+Bxx=C=Cxx only works when only works when
X=2X=2
Pythagorean Pythagorean TheoremTheorem
AA22+B+B22=C=C22
Michelle MoardMichelle Moard
Resources Used Resources Used & other good sites& other good sites
Math 128 Modern GeometryMath 128 Modern Geometry LinkLink
Dr. Peggie HouseDr. Peggie House Pythagorean TheoremPythagorean Theorem LinkLink
Pythagorean Theorem AppletPythagorean Theorem Applet LinkLink
PythagorasPythagoras LinkLink
WikipediaWikipedia LinkLink