Shearing a figure preserves its area. Use the green rectangles to reason why [math]\triangle ABD[/math] has the same area as [math]\triangle ABC[/math] for any position of [math]D[/math] along that line, as long as [math]A[/math], [math]B[/math], and [math]C[/math] are fixed. [br][br]What if you begin with a different [math]\triangle ABC[/math]?
How does the area of [math]\triangle ABD[/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 [math]\triangle ABD[/math]. To see this, cut off the parts of the rectangles that are outside the triangle, 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]D[/math] is moved? Explain.
The rectangles are translated parallel to the base, segment [math]AB[/math]. Their lengths, heights, and areas stay the same.
Move points [math]A[/math], [math]B[/math], and [math]C[/math] to check that your conclusions and explanations hold for a different [math]\triangle ABC[/math].[br][br]And are we really sure that the lengths of the rectangles stay the same when [math]D[/math] is moved? Moving [math]D[/math] creates an optical illusion suggesting the lengths might change. We resolve this in the next sketch.