Dividing a Hexagon into 8 Equal Parts

Date: 06/11/99 at 02:02:34
From: Stef
Subject: Dividing a hexagon into 8 equal parts

How do you divide a hexagon into 8 equal parts? I figured out how to 
divide it into 4 equal trapezoids but I have to figure out how to 
divide it into 8 equal parts.

Date: 06/11/99 at 05:56:39
From: Doctor Pete
Subject: Re: Dividing a hexagon into 8 equal parts

Hi,

I assume by "hexagon" you mean "regular hexagon," with all sides equal 
and all interior angles equal. A novel way to dissect it into 8 
congruent parts is as follows:

Suppose you have the vertices labeled A, B, C, D, E and F in clockwise 
order. Let O be the center of the hexagon ABCDEF. Draw regular hexagon 
A'B'C'D'E'F' such that it also has center O, and A'B' is parallel to 
AB, B'C' is parallel to BC, etc., and furthermore, A'D' is half of AD. 
Then draw lines AA', BB', CC', DD', EE', FF', and draw line A'D'. This 
divides the original hexagon into 8 congruent trapezoids.

  

Now, here's a question for you: How many different ways can you divide 
the same hexagon up into 8 congruent trapezoids like the ones you 
found in the above example? (Hint: Take the trapezoids you found in 
the example and divide each up into three equilateral triangles.)

- Doctor Pete, The Math Forum
  http://mathforum.org/dr.math/