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

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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Proof AProof A

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A2+B2=C2

C

AB

B2+C2=A2

A2=B2+C2

C

B

A

B2=A2+C2

C

B

A

Proof BProof B

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B

C

We start with a right triangle.

Proof BProof B

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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

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B

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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

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B

C

A

B

C

The area of the small square is C x C

or C2

Proof BProof B

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B

C

A

B

C

The area of the original triangle is

½ (A x B)

Proof BProof B

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B

C

A

B

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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

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B

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A

B

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(A+B)2 = 4 (½ (A x B)) + C2

Proof BProof B

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B

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A

B

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(A+B)2 = 4 (½ (A x B)) + C2

Proof BProof B

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B

C

A

B

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(A+B)2 = 4 (½ (A x B)) + C2

Proof BProof B

A

B

C

A

B

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(A+B)2 = 4 (½ (A x B)) + C2

Proof BProof B

A

B

C

A

B

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(A+B)2 = 4 (½ (A x B)) + C2

Proof BProof B

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B

C

A

B

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(A+B)2 = 4 (½ (A x B)) + C2

Proof BProof B

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B

C

A

B

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(A+B)2 = 2AB + C2

Proof BProof B

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B

C

A

B

C

(A+B)2 = 2AB + C2

Proof BProof B

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B

C

A

B

C

A2+2AB+B2 = 2AB + C2

Proof BProof B

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B

C

A

B

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A2+2AB+B2 = 2AB + C2

Proof BProof B

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B

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A

B

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A2+2AB+B2 = 2AB + C2

Proof BProof B

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B

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A

B

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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