Notice the Continuation Arrow at the top of the microworld. That means that you should click the arrow before you leave this page to continue the experiment. Once you build the quadrilateral here, you will be able to experiment with it if you click the continuation arrow. When the next "room" of the lab appears, it will have a "return arrow" so that you can come back to this "room" if you want to tinker with things. You can move back and forth between these rooms of the microworld without leaving the page. But if you do leave the page (say, by pressing the Back button, or clicking one of the "pointing fingers" at the bottom of the page), then you will have to start over.

 

  To build your quadrilateral, click at 4 lengths in the blue ruler window on the bottom of the screen.

 

This determines the lengths of the four sides:  Blue, Red, Green, and Yellow, in that order.  The lengths will be rounded to integers unless you select "tenths" in the precision field, in which case the lengths will be rounded to tenths.  Toggle back and forth between integer and tenths precision by pushing the precision button.

 

Once you have selected the 4 lengths, the segments will be drawn in the window, and the lengths will be printed.   Next, you may select the angle between the blue and red segments.  This will almost determine the quadrilateral.  (Why almost?)   You select that angle by dragging the protractor meter to the angle you want. Just click in the protractor window, holding the left mouse button down, and move the meter to the desired angle. 

 

 

The current angle will be printed, and it will be selected using the "precision" that you have set.  When you are satisfied, release the left mouse button, and press the right mouse button.

 

       After the quadrilateral is built, it will be remembered.  Then, stay on this page, but press the Continue Arrow at the top of the Microworld to experiment with it.   On that story page, we will experiment a little with the quadrilaterals that can be formed from the one just created.   The basic picture is this:

 

 

 

and we will refer to its labels.  T is the angle between the red and blue segments, and P is the angle between the green and yellow segments.  We will generally measure these angles in radians.  So approximately 1.57 units corresponds to 90 degrees etc.

       

Now our objective here is twofold.  First, we would like to see how the angles T and P vary as the quadrilateral is deformed.  Next, we want to look at how the area of the quadrilateral changes.

 

         To "deform" the quadrilateral, do this.  First click with the left mouse button on a vertex of the quadrilateral.  A blue  "sprite" should appear there to indicate that you have selected it.  Next, hold the left mouse button down and drag the sprite to a new position.  When finished, release the left mouse button and press the right mouse button.  This indicates that you are done. 

 

          After you do these things, the labeled vertex will jump to the new position.  The vertex opposite that vertex will stay fixed.  The other two vertices will move in the way they must to produce a new quadrilateral with the sides keeping their given lengths.  Those lengths, by the way, are printed with their color labels (as in the diagram above) in the field:

 

 

After you do this you will receive two pieces of information  about the new quadrilateral.  Its area is printed in the field just below the "T versus P curve"  window:

 

 

This area is calculated using the formula

 

           

 

     And a point is plotted in that window.  The point represents the two angles, T and P shown in the picture above.  They are measured in radians, with T on the horizontal axis and P on the vertical axis.  To help you see what is going on, there is a button that draws the curve expressing the relation between T and P for various quadrilaterals.  That curve is given by equation (2.2) that we saw earlier:

 

 

            As you change the shape of the quadrilateral, you should notice the new points being deposited on that curve.  Different sets of lengths: a,b,c, and d give different curves of course.

 

           Now here are a few questions.

 

       Exercise:  Construct a rectangle.  Why do the points (T,P) lie in a straight line?  What is that line? 

 

       Exercise:   For the four lengths determining a rectangle, notice that deformations give parallelograms.  Which of these has the largest area?  If the lengths you chose are reordered, will you always get a parallelogram?

 

                          Describe another shape that you can get and say when the area of this shape is maximized. 

                          How does the maximum area of the family of parallelograms compare with the maximum area of this new family?  Why?

 

            Once you are satisfied that you understand these things, you will be ready to continue our story by moving on to the next page.