1b+-+Euclid+(ca.+300+BC)

"[Euclid's] system was studied by Greek and Roman scholars for a thousand years, then translated into Arabic around 800 A.D. and studied by Arab scholars, too. It became the standard for logical thinking throughout medieval Europe. It has been printed in more than 2000 different editions since it first appeared as a typeset book in the 15 century. That system is Euclid's description of plane geometry ( Berlinghoff & Gouvea, 2004, p. 155)." "The name Euclid until quite recently was practically synonymous with elementary school geometry (Bell, 1937, p. 299)."

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Created using GeoGebra 3.2.40.0 GeoGebra - Dynamic Mathematics for Everyone []

1. The yellow square has as one of its sides the hypotenuse of a right triangle. Label that length, c. Label the other squares a and b, respectively.

2. Move the "Sizes" slider to verify this configuration holds for all sizes of squares. (Note what happens when a square has zero length.)

3. With the "Sizes" slider adjusted so that the blue and green squares are not of zero size, alternately slide the "Left" and "Right" slider. What happens? Does this suggest a proof? Can you, at this point, prove the Pythagorean Theorem from this drawing?

4. Check the "Show Hints" checkbox. Now, slowly slide the "Left" slider until the right vertex turns from red to yellow. Compare the area of the empty square to that of the blue parallelogram? (Hint: Use the red segments to establish length and width or base and height). Can you prove a relationship? What is it?

5. Continue sliding the "Left" slider until the vertex point touches the yellow point on the top of the yellow square. Note the dashed boundaries where the parallelogram used to be. Compare the area of that parallelogram to the blue one. Again, use the red segments to help you determine the area.

6. Slide the "Left" slider until the blue area is completely within the yellow square. How does the area of the original square compare with the area now in blue?

7. Repeat steps 4 through 6 for the other side. (Replace "Left" with "Right" and "blue" with "green" as appropriate.)

8. Use what you discovered to formalize a proof based on this interactive construction.