On Wed, 28 Sep 2005 05:01:12 GMT, charleswehner@[EMAIL PROTECTED]
wrote:
>You know - you create a square on side 3, add the square on side 4 and
>get the square on side 5. This is all to do with AREAS.
>
>However, what if you examine the PERIMETER of the square on side 3, and
>compare it with the perimeter of the triangle?
The perimeter of a square drawn on the side is four times the length
of that side?
You'd be better perhaps to consider still areas: Areas of *any*
similar figures drawn similarly on the sides of a right triangle bear
the same Pythagorean relation****p as do the squares; the sum of the
two smaller add to the larger. Also, it is useful to know that in
similar figures linear measures are in the same ratio as the square
roots of the areas. That can be useful in so many ways.
So, you can draw similar triangles, or semi-circles, or whatever [even
very odd-looking, but exactly similar closed shapes] on the sides of a
right triangle, and the sum of the areas of the smaller will add to
the larger. Then there will also be a relation****p between the
perimeters since the sides of the triangle are each the same factor of
the total perimeter of that similar figure. For example, the side of
the triangle is a diameter of a semi-circle drawn on that side, and is
so 2:(2+Pi) of that shape's perimeter, large or small.
I'd be careful of what I'd teach to children and what I'd leave for
later examination in a fuller context. Children have at first enough
difficulty with the normal Pythagorean theorm as it is usually taught.
Some discoveries are exciting to you because you already have the
necessary background to compare and contrast. To them, it could be
something the teacher wants, so they will do it, but it might have
little or no meaning to them out of a larger context.
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