When I taught a geometry methods course at City College last fall, I devoted an entire class to investigating area. We focused on problems where triangles were sheared, transforming into new triangles, but maintaining their area. The two Web Sketchpad activities that follow introduce shearing and present a problem with a surprising result that can be explained using properties of shearing.

Let’s start with the basics. Follow these steps with the sketch below (and here). For help, this YouTube video walks you through the construction.

• Construct an arbitrary triangle ABC.
• Construct an altitude of ABC, using side BC as the base.
• Drag point A. Does your altitude exist regardless of where you drag the point? If not, rebuild the altitude so that it is more robust.
• Calculate the area of your triangle. Your calculation should automatically update when you drag the triangle’s vertices.
• Drag point A. As you do, try to keep the area measurement nearly constant. Now do the same, but trace point A. What do you notice about the path of point A?
• Construct a line parallel to side BC of your triangle. Attach point A to the line. What do you notice about the area of your triangle as you drag point A along the line?

In the sketch below (and here) ABCD and ECFG are adjoining squares. As you drag point E, notice that, rather surprisingly, the area of ∆BDG does not change. And if you drag point E very close to point C, you can see that the area of ∆BDG looks to be half the area of square ABCD. How interesting!

Use the Trace widget to trace point G as you drag point E. What do you notice about point G‘s path and how can this help to explain your observations of ∆BDG‘s area?

On page 2 of the websketch, you can build this construction yourself, starting from scratch.