Overview of Section 3.2 The Pythagorean Theorem

An equivalent form of the Pythagorean Theorem is:

The area of a square constructed on the hypotenuse of a right triangle equals the sum of the areas of the squares constructed on the sides of the triangle.

Who was Pythagoras? What is known about him? Look at the the entry on the History of Mathematics web site at St. Andrews.

A reference: The book by Loomis:

E. S. Loomis, The Pythagorean Proposition, NCTM, 1968

has 357 different proofs of the Pythagorean theorem but it is no sense 'complete.' Loomis wrote the book around 1900 and it was republished by NCTM as a Classics in Mathematics book in 1968. At the time he wrote the book Loomis claimed he selected his 357 proofs from over 1400. His count may be a little inflated by with variations of algebraic manipulations derived from the same diagram for a proof.

On the Web. There are hundreds of good sites. Do a Google search. Or, I recommend

http://www.cut-the-knot.org/pythagoras/index.shtml#7

where there are 79 proofs fully presented with diagrams, animations, and links to much more. There are proofs that are essentially algebraic, others that are essentially geometric, but none that are trigonometric . . . Why so?

We will explore several different proofs of this theorem.

Proof 1 from Libeskind -- a dissection model of a square a + b on a side where a and b are legs of a right triangle

Open GSP file

Proof 2 from Libeskind -- an algebraic proof based on the above diagram. See the GSP file.

A proof using shear transformations. Open GSP file.

The basic idea is that under a shear transformation the area remains invariant. So under the shear, square ADEB becomes parallelogram ADEB; square BHIC becomes parallelogram BHIC. The line JK divides square AGFC into two rectangles and under the shear rectangle AGJK become s parallelogram AGJK and rectangle CFJK becomes parallelogram CFJK. In the final position with shears relative to the line of the altitude JK, parallelogram ADEB is congruent to parallelogram AGJK and parallelogram BHIC is congruent to parallelogram CFJK.

SEE THE ANIMATIONS IN THE GSP FILE.

EUCLID's PROOF

The equality of the areas can be shown by shear transformations of the respective triangles. For example shear the yellow triangle along AC to A and the orange triangle along the line of the altitude to C. The resulting triangles are congruent.

Try this to see an animation.

http://www.cut-the-knot.org/pythagoras/Pyth69PWW.shtml

# Problems (worked in the text)

A. Construct any segment one unit long, and use the relationship to construct a segment of length

Open GSP File

B. Find the area of an equilateral triangle with side length a.

Open GSP File Note: There are two derivations on the GSP file. The first uses the right triangle formed by an altitude and 1/2 of the equilateral triangle. The second uses Heron's formula (see item C below).

C. Heron's formula for the area of a triangle. See http://jwilson.coe.uga.edu/emt725/Heron/Heron1.html

D. Impossible to construct an equilateral triangle on a geoboard.

E.

Consider this: