Shearing
Shearing a figure preserves its area. Use the green rectangles to reason why has the same area as for any position of along that line, as long as , , and are fixed.
What if you begin with a different ?
How does the area of compare to the sum of the areas of the green rectangles when or ? What about ? Explain.
What happens to the rectangles (and their areas) when is moved? Explain.
Move points , , and to check that your conclusions and explanations hold for a different .
And are we really sure that the lengths of the rectangles stay the same when is moved? Moving creates an optical illusion suggesting the lengths might change. We resolve this in the next sketch.