Transcript Proof of the Pythagorean Theorem

Pythagorean Theorem
by Bobby Stecher
[email protected]
The Pythagorean Theorem as some
students see it.
c
a
b
2
2
2
a +b =c
A better way
2
c
2
a
a
c
2
2
2
a +b =c
b2
b
Pythagorean Triples
(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)
Pythagorean Triples
Pythagorean Triples
http://www.cut-the-knot.org/Curriculum/Algebra/PythTripleCalculator.shtml
The distance formula.
(x1,y1)
c = distance
b = y2-y1
a = x2-x1
(x2,y2)
The Pythagorean Theorem is often easier for students to learn
than the distance formula.
Proof of the Pythagorean Theorem from
Euclid
Euclid’s Proposition I.47 from
Euclid’s Elements.
Proof of the Pythagorean Theorem
Line segment CN is
perpendicular to AB and
segment CM is an altitude of
ΔABC.
Proof of the Pythagorean Theorem
Triangle ΔAHB has base AH
and height AC.
Area of the triangle ΔAHB is
half of the area of the square
with the sides AH and AC.
Proof of the Pythagorean Theorem
Triangle ΔACG has base AG
and height AM.
Area of the triangle ΔACG is
half of the area of the rectangle
AMNG.
Proof of the Pythagorean Theorem
Recall
AG
is equal
that ΔACG
to AB because
is half of both
are sides of
rectangle
AMNG
the same
and ΔAHB
square.
is
equal to half of square ACKH.
AC is equal to AH because both
Thus
square
ACKH
is square.
equal
are sides
of the
same
to rectangle AMNG.
Angle <CAG is equal to <HAB.
Both angles are formed by
adding the angle <CAB to a
right angle.
ΔACG is equal ΔAHB by SAS.
Proof of the Pythagorean Theorem
Triangle Δ MBE has base BE
and height BC
Triangle Δ MBE is equal to half
the area of square BCDE.
Proof of the Pythagorean Theorem
Triangle Δ CBF has base BF
and height BM.
Triangle Δ CBF is equal to half
the area of rectangle BMNF.
Proof of the Pythagorean Theorem
BE
is square
equal toBCDE
BC because
Thus
is equalboth
are
sides of the
same square.
to rectangle
BMNF.
BA is equal to BF because both
are sides of the same square.
Angle <EBA is equal to <CBF.
Both angles are formed by
adding the angle <ABC to a
right angle.
ΔABE is equal ΔFBC by SAS.
World Wide Web java applet for
Euclid’s proof.
http://www.ies.co.jp/math/java/geo/pythafv/y
hafv.html
Additional Proofs of the
Pythagorean Theorem.
Proof by former president James Garfield.
http://jwilson.coe.uga.edu/emt669/Student.Folders/Huberty.Gr
eg/Pythagorean.html
More than 70 more proofs.
http://www.cut-the-knot.org/pythagoras/
A simple hands on proof for
students.
Step 1: Cut four identical right triangles from a piece of paper.
c
a
b
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
b
c
c
a
b
Area of each part
4 Triangles
= 4 x (ab/2)
1 Red Square = c2
(a + b)2 = 2ab +c2
a2 + 2ab + b2 = 2ab +c2
a 2 + b2 = c 2
c
b
a
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
a–b
(a – b)2 + 2ab +c2
c
a–b
a2 – 2ab + b2 + 2ab = c2
a2 + b2 = c 2
c
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 < c2, then triangle ABC is an obtuse triangle.
If a2 + b2 > c2, then triangle ABC is an acute triangle.
Cartesian equation of a circle.
x2 + y2 = r2 is the equation of a circle with the center at
origin.
Pythagorean Fractal Tree
Students can create a
fractal using similar right
triangles and squares.
Using right triangles to
calculate and construct
square roots.
Was Pythagoras a square?
The sum of the area’s 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.
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.
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/pythagoras.htm
http://www.cut-the-knot.org/
http://www.contracosta.edu/math/pythagoras.htm
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.html
• http://jwilson.coe.uga.edu/emt669/Student.Folders/Hu
berty.Greg/Pythagorean.html
• http://www.cut-the-knot.org/pythagoras/