Pythagorean Theorem Laboratory

Write your answers in a Word document that you will print out and turn in.  USE COMPLETE SENTENCES.  After completing this lab you should have a thorough understanding of both geometric and algebraic implications of the Pythagorean Theorem, and you will have worked through “sketches” of a few proofs that have developed over the centuries.  Visit Proofs 1-3, the “Exposition” and “Biography,” and at least one of the remaining “proofs” (your choice).  More proofs = better grade.

Proof 1:  Go to  http://www.cut-the-knot.org/pythagoras/index.shtml ; READ down through “Remark 1,” then read and work through the “first proof” below the remarks.

  1. Which culture is credited with recognizing the Pythagorean Theorem over 3000 years ago?
  2. Which formula we have studied recently is an important part of this proof?  How is it used?

Proof 2:  Next, go to http://oneweb.utc.edu/~Christopher-Mawata/geom/geom7.htm and work through the proof.

  1. How do you know the pink shape surrounded by four triangles is a square, instead of a rhombus?

  2. In the final “scene” of the proof, how do you know that the two pink shapes are squares?

Proof 3:  Go to http://www.cut-the-knot.org/Curriculum/Geometry/ArrangePyth.shtml.  Drag the bottom corner of the red square to make the image change.

  1. The drawing on the left consists of 4 yellow triangles and a red square; if the triangles have the usual “a-b-c” dimensions, what is the area of the red square?  What is the area of all four yellow triangles?  What are the dimensions (length and width) of the combined figure?
  2. The drawing on the right consists of the same 4 yellow triangles, arranged in rectangles, with two red squares.  What are the areas of the two red squares?  What is the area of all four yellow triangles?  What are the dimensions (length and width) of the combined figure?
Exposition:  Scan http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html (an electronic annotated version of Euclid’s elements).  The proof follows “Proof 1” above, but justifies every step using the angle addition postulate and others (some of which we have not yet studied).  Some of the proof may be confusing.  Question:  Identify another culture that “discovered” Pythagoras’ theorem.  Identify a theorem, property, fact, etc. in the proof that we have not yet proven.

Biography:  Go to http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html and scan the biography of Pythagoras.

  1. About how long did Pythagoras live?
  2. Scroll down to the bottom, click on “Some Quotations,” and copy your favorite of those listed.
  3. Identify one “interesting” fact about Pythagoras’ life.

Visit at least one of the following:

Proof 4:  Go to http://www.ies.co.jp/math/java/geo/pitha1/pitha1.html.  For this one, your goal is to move the pink and green squares down into the large empty square (applet at bottom of screen). 

  1. Note that this Japanese site labels the triangle differently than we’re used to… based on its labeling, how is the theorem expressed?
  2. Were you successful?  If so, tell how you did it (one square is enough… the other works the same way).  If not, go to http://www.uni.uiuc.edu/~hcrussel/hint.html for a hint and try again.  If you still don’t succeed, write a few sentences explaining what difficulty you had.
  3. This proof uses a modification of the formula you identified under Proof 1.  What is the modification?

 

Proof 5:  Go to http://www.ies.co.jp/math/products/geo2/applets/pythasvn/pythasvn.html. 

Question:  Explain what this “proof” is doing.  I wrote “proof” in quotation marks because it is not clear that this demonstration truly constitutes a proof… either explain why it does, or tell what is wrong with it.

 

Proof 6:  Go to "Ask Dr. Math" (from the Math Forum at Drexel University) for a description of a proof proposed by President Garfield.  You can see a better graphic (and all the calculations involved) at http://mathworld.wolfram.com/PythagoreanTheorem.html; scroll down to the graphic and paragraph after Equation 18.

Question:  What new kind of quadrilateral is involved in the proof?  Based on the proof, give a good definition of this new quadrilateral.  Try to explain how the area formula works for this one.

 

Proof n:  Go to http://www.cut-the-knot.org/pythagoras/index.shtml and choose any of the proofs not done above.

Question:  Describe the proof in your own words.  Tell why you chose it.