I did enjoy trying to figure out how to balance the different types of quadrilaterals.  The first method that I used was guessing.  I would simply put the object on the end of my eraser to see if it would balance.  Then I did what we all did, and tried to split the quad into two triangles.  So I did that, found the two centroids, and made line that connected the two.  I later found out that this was called the balancing line.  I could balance the quad with my ruler, but to balance it on my eraser took a little more quessing, and I was not satisfied with this.  Then Inchul gave me a little hint, and asked me if those two triangles that I created were the only triangles that could be formed from that particular quad.  So next I drew a line down the other diagonal, creating two new triangles, and again found the two cenroids, and connected them with a line.  Then I noticed that the two balancing lines intersected.  So I put my eraser on that point, and it balanced.  This was the process that I went through:

These were my original quads.  They are the same, they are just split down the two diagonals.
 
 

Then I found the centroids of all four triangles.  And connected them.

Then I overlaped the two on the computer, but in class I was simply drawing on the cardboard.  Then were the two balancing lines crossed that was my balacing point B.

    I was able to conclude that you can find the balancing point of any quad, not matter how irregular it is.  By spliting the quad into triangles and finding the centroids of each, we found the balancing point of each triangle.  Meaning that the mass of the triangles are evenly distributed around that point.  then when connected the two centroids, we found one of the balancing lines of that quad.  Meaning that on either side of that line, the mass of the quad was evenly disributed.  Then by repeating the same steps, down the other diagonal we found another balacing line, again were the mass was equal on either side of that new line.  In our minds we knew that somewhere on that balacing line the quad would be equally balanced.  Then when we found that the other balancing crossed the original, we knew that was the ponit on the balancing line that was the center of mass for that quad.

We even tried some concave quads, and found that it did have a balancing line, it was just that it was not inside the quad.

To see a script on how the center of mass is formed click here.  You can also see how this script works if you go into GSP with a new scetch, create four points, highlight them in a counterclockwise fashion, and then play the script.  If you play the script step by step, you can see how the centroids are made, and then connected.