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In 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".
There are more than 300 different proofs!!
We can generalize to three dimensions for a parallelopiped
(prism):
Here are some applications of the Pythagorean theorem. Can you see how to
prove
them?
Suppose
and
are integers such that
Give an argument that if one of the integers is odd, then exactly one of the
other integers is odd.
Use the Pythagorean theorem to find the area of an equilateral triangle of
side length
.
(Hint. Divide the equilateral triangle into two equal right triangles. You
know one leg and the hypotenuse.)
Make exactly two cuts of the (left) figure below to form the three pieces that
can be rearranged into a perfect
square.
Let
the area of a hexagon inscribed in a circle, and
be the circumscribed hexagon. If
represents the area of the inscribed 12-gon (i.e. twelve equal sides), show
that
Note, this formula was used by Archimedes in about 250BC to make his famous
estimation of
You recall
[See
picture above.]
The golden section is shown
below:
Show
that if
,
then
The golden section is used to construct the regular pentagon.
Cut the figure below with ONE straight cut that passes though the point
so that the figure is cut into exactly two parts of equal
area.
You have a
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.