Shearing a figure preserves its area. Use the green rectangles to reason why parallelogram [math]ABEF[/math] has the same area as parallelogram [math]ABCD[/math] for any position of [math]F[/math] along that line, as long as [math]A[/math], [math]B[/math], and [math]C[/math] and [math]D[/math] are fixed. [br][br]What if you begin with a different parallelogram [math]ABCD[/math]?
How does the area of parallelogram [math]ABCD[/math] compare to the sum of the areas of the green rectangles when [math]n=2[/math] or [math]n=3[/math]? What about [math]n=20[/math]? Explain.
For any number of rectangles, the sum of their areas is equal to the area of parallelogram [math]ABCD[/math]. To see this, cut off the parts of the rectangles that are outside the parallelogram, and move them to fill in (exactly) the "missing" pieces of area inside the triangle.
What happens to the rectangles (and their areas) when [math]F[/math] is moved? Explain.
The rectangles are translated parallel to the base, segment [math]AB[/math]. Their lengths, heights, and areas stay the same.