Re: math league problem

Duncan (dchiu@mail.idt.net)
Thu, 12 Dec 1996 02:01:09 -0800

Chuck Biehl wrote:
>
> We have this problem that neither of us can figure out. I can't include
> the diagram, so try to use your imagination.
>
> Diagram: A scalene triangle with horizontal base. Vertices are labeled
> clockwise from the bottom left as A, B, and C. Horizontal segments are
> drawn in the triangle parallel to AC such that the altitude is divided
> into 5 equal segments, making the triangle divided into 5 "strips" of
> equal width. The second strip from the bottom is shaded.
>
> The question: "What is the ratio of the shaded area to the unshaded area?"
> The answer is given as 7/18, with an explanation that we cannot
> understand. I won't share the explanation here so as to not spoil the fun.
> Can somebody help us out?
>
> L. Charles (Chuck) Biehl
> The Charter School of Wilmington
> Wilmington, DE 19807
> (v)302.651.2727 (f)302.652.1246
> http://dimacs.rutgers.edu/~biehl

Chuck,
as follows:
Triangle 1 has altitude of 1 unit --- the top one;
Triangle 2 has altitude of 2 units --- the top one of 2 "strips";
3 3 3
4 4 4
Triangle 5 5 5

Put them side by side to see the ratio of altitudes:

1:2:3:4:5

Their area ratio are:
1x1 : 2x2 : 3x3 : 4x4 : 5x5 (Similar triangles)

Shaded strip area = 4x4 - 3x3 = 7 (Triangles 4 - 3)
Unshaded area = 5x5 - 7 = 18 (Whole triangle - shaded)