Pythagorean Theorem

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Pythagorean Theorem

• by Bobby Stecher
• mark.stecher_at_maconstate.edu

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The Pythagorean Theorem as some students see it.
a2b2c2
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A better way
c2
c
a2
a
a2b2c2
b
b2
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Pythagorean Triples
A few observations 1. One of the legs of the
right triangle is a multiple of 3. 2. One of the
legs of the right triangle is a multiple of 4. 3.
One of the three is a multiple of 5.
(3,4,5) (5,12,13) (7,24,25) (8,15,17) (9,40,41)
(11,60,61) (12,35,37) (13,84,85) (16,63,65)
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Pythagorean Triples
Although the Pythagorean Theorem was not known,
the Pythagorean triples were familiar to the
Babylonians. (Livio, 28)
Babylonians discovered that Pythagorean triples
can be constructed using the following method.
1. Choose any two whole numbers p and q. Let q
be the smaller number.
2. Compute p2 q2 , 2pq, and p2 q2
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Pythagorean Triples
1. Choose any two whole numbers p and q. Let q
be the smaller number.
Let p 2 and q 1.
2. Compute p2 q2 , 2pq, and p2 q2
p2 q2 4 1 3 2pq 2(2)(1) 4 p2 q2
4 1 5
http//www.cut-the-knot.org/Curriculum/Algebra/Pyt
hTripleCalculator.shtml
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The distance formula.

(x1,y1)
c distance
b y2-y1
(x2,y2)
a x2-x1
The Pythagorean Theorem is often easier for
students to learn than the distance formula.
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Proof of the Pythagorean Theorem from Euclid
Euclids Proposition I.47 from Euclids Elements.
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Proof of the Pythagorean Theorem
Line segment CN is perpendicular to AB and
segment CM is an altitude of ?ABC.
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Proof of the Pythagorean Theorem
Triangle ?AHB has base AH and height AC.
Area of the triangle ?AHB is half of the area of
the square with the sides AH and AC.
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Proof of the Pythagorean Theorem
Triangle ?ACG has base AG and height AM.
Area of the triangle ?ACG is half of the area of
the rectangle AMNG.
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Proof of the Pythagorean Theorem
AG is equal to AB because both are sides of the
same square.
Recall that ?ACG is half of rectangle AMNG and
?AHB is equal to half of square ACKH.
AC is equal to AH because both are sides of the
same square.
Thus square ACKH is equal to rectangle AMNG.
Angle ltCAG is equal to ltHAB. Both angles are
formed by adding the angle ltCAB to a right angle.
?ACG is equal ?AHB by SAS.
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Proof of the Pythagorean Theorem
Triangle ? MBE has base BE and height BC
Triangle ? MBE is equal to half the area of
square BCDE.
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Proof of the Pythagorean Theorem
Triangle ? CBF has base BF and height BM.
Triangle ? CBF is equal to half the area of
rectangle BMNF.
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Proof of the Pythagorean Theorem
BE is equal to BC because both are sides of the
same square.
Thus square BCDE is equal to rectangle BMNF.
BA is equal to BF because both are sides of the
same square.
Angle ltEBA is equal to ltCBF. Both angles are
formed by adding the angle ltABC to a right angle.
?ABE is equal ?FBC by SAS.
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World Wide Web java applet for Euclids proof.

• http//www.ies.co.jp/math/java/geo/pythafv/yhafv.h
  tml

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Additional Proofs of the Pythagorean Theorem.
Proof by former president James Garfield.
http//jwilson.coe.uga.edu/emt669/Student.Folders
/Huberty.Greg/Pythagorean.html
More than 70 more proofs. http//www.cut-the-knot.
org/pythagoras/
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A simple hands on proof for students.
Step 1 Cut four identical right triangles from
a piece of paper.
c
a
b
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A simple hands on proof for students.
Step 2 Arrange the triangles with the
hypotenuse of each forming a square.
a
b
Area of large square (a b)2
a
c
Area of each part 4 Triangles 4 x (ab/2) 1
Red Square c2
b
c
(a b)2 2ab c2
c
b
a2 2ab b2 2ab c2
c
a2 b2 c2
a
b
a
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Alternate arrangement
Area of large square c2
c
Area of each part 4 Triangles 4 x
(ab/2) 1 Purple Square (a b)2
b
a
a b
c
(a b)2 2ab c2
a b
a b
c
a2 2ab b2 2ab c2
a b
a2 b2 c2
c
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The converse of the Pythagorean Theorem can be
used to categorize triangles.
If a2 b2 c2, then triangle ABC is a right
triangle.
If a2 b2 lt c2, then triangle ABC is an obtuse
triangle.
If a2 b2 gt c2, then triangle ABC is an acute
triangle.
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Cartesian equation of a circle.
x2 y2 r2 is the equation of a circle with
the center at origin.
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Pythagorean Fractal Tree
Students can create a fractal using similar right
triangles and squares.
Using right triangles to calculate and construct
square roots.
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Was Pythagoras a square?
The sum of the areas of the two semi circles on
each leg equal to the area of the semi circle on
the hypotenuse. The sum of the areas of the
equilateral triangles on the legs are equal to
the area of the equilateral triangle on the
hypotenuse.
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Extensions and Ideas for lessons

• Does the theorem work for all similar polygons?
  Is there a trapezoidal version of the Pythagorean
  Theorem?
• Using puzzles to prove the Pythagorean Theorem.
• Make Pythagorean trees.
• Cut out triangles and glue to poster board to
  demonstrate a proof of Pythagorean Theorem.
• Create a list of Pythagorean triples and apply
  proofs to specific triples.
• Use Pythagorean Theorem with the special right
  triangles.
• Categorize triangles with converse theorem.

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References
Boyer, Carl B. and Merzbach, Uta C. A History of
Mathematics 2nd ed. New York John Wiley Sons,
1968. Burger, Edward B. and Starbird, Michael.
Coincidences, Chaos, and All That Math Jazz. New
York W.W. Norton Company, 2005. Gullberg,
Jan. Mathematics From the Birth of Numbers. New
York W.W. Norton Company, 1997. Livio,
Mario. The Golden Ratio. New York Broadway
Books, 2002. http//www.contracosta.edu/math/pyth
agoras.htm http//www.cut-the-knot.org/
http//www.contracosta.edu/math/pythagoras.htm

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Links

• http//www.contracosta.edu/math/pythagoras.htm
• http//www.cut-the-knot.org/
• http//www.contracosta.edu/math/pythagoras.htm
• http//www.ies.co.jp/math/java/geo/pythafv/yhafv.h
  tml
• http//jwilson.coe.uga.edu/emt669/Student.Folders/
  Huberty.Greg/Pythagorean.html
• http//www.cut-the-knot.org/pythagoras/