Slanty Squares

I attended a well-crafted session with Kit Luce yesterday. She handed out a square and asked us what questions it brought to mind. One participant wondered if the green square was half of the total. I thought that was interesting and also thought that the green area was clearly more than a half.

I took the rather woeful photo above, imported it into the mathies Notepad Tool, and used a grid to estimate the areas.  I was surprised that the green area seemed to be significantly less that one-half.

I then turned to Sketchpad to model the situation.  Apparently, Blogger doesn't let me use an iframe to embed a web sketch at a regular http:// address, so you have to visit the interactive sketch in its own window.

What is special about the place where the green square is exactly one-half of the area?

Do you find this as counter-intuitive as I do?

An Interactive Version of the Triangle Investigation

In the previous post, I captured movies of my investigation with The Geometer's Sketchpad. Web Sketchpad allows for including an HTML 5 version of the sketch on a webpage, like this one.

Drag the yellow dot, currently on the triangle to trace out the various positions and area of the triangle.

 

Note that I have enforced the maximum side length of 10 in a very strange way. Can you describe what I have done? Can you do it in a similar or better way?

Where do you have to place the yellow dot in order to have an area of 0?

How can you drag the yellow dot to keep the area the same?

What other patterns do you see in the behaviour? (There are lots more in the previous post)