The Pythagorean what???

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wizard__4.pngIn the movie The Wizard of Oz, Dorothy (Judy Garland) meets up with the Lion, the Tin man, and the Scarecrow. Each wants a gift from the great Wizard of Oz. The Scarecrow wants a brain. When he finally gets one, he pronounces .... "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side."

What??? What the scarecrow meant was the statement of the famous Pythagorean Theorem:

"The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse".


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There are more than 300 different proofs!!

We can generalize to three dimensions for a parallelopiped (prism):
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Here are some applications of the Pythagorean theorem. Can you see how to prove them?
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wizard__21.pngNow its your turn. Use the Pythagorean theorem, if you can.

  1. Suppose $a,\ b,\ $ and $c$ are integers such that $a^{2}+b^{2}=c^{2}.$ Give an argument that if one of the integers is odd, then exactly one of the other integers is odd.

  2. Use the Pythagorean theorem to find the area of an equilateral triangle of side length $a$. (Hint. Divide the equilateral triangle into two equal right triangles. You know one leg and the hypotenuse.)

  3. Make exactly two cuts of the (left) figure below to form the three pieces that can be rearranged into a perfect square.
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  4. Let $a$ the area of a hexagon inscribed in a circle, and $A$ be the circumscribed hexagon. If $t$ represents the area of the inscribed 12-gon (i.e. twelve equal sides), show that $t^{2}=aA.$ Note, this formula was used by Archimedes in about 250BC to make his famous estimation of $\pi .$ You recall MATH[See picture above.]

  5. The golden section is shown below:
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    Show that if $AC~=1$, then MATH The golden section is used to construct the regular pentagon.

  6. Cut the figure below with ONE straight cut that passes though the point $A$ so that the figure is cut into exactly two parts of equal area.
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  7. You have a $1\times 3$ strip of paper. Show how to cut this strip, making only straight line cuts, and then arrange the paper into a cube, with no overlays and no gaps.
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