Readers of the blog have come to know and love hearing about the algebra whiz kids from East Tipp Middle School in Indiana (see here and here).
They have just solved the most recent problem I gave them in all sorts of ingenious ways. Here was the challenge:

We all know that the Pythagorean Theorem states that A² + B² = C² in right-angled triangles in Euclidean geometry.

But you may have forgotten that the “squared” is meant literally here: the square on side A (i.e. the area of that square) + the square on side B ((i.e. the area of that square) = the square on side C, i.e. the area of that square.

QUESTION: can the theorem be generalized to shapes other than squares?

For example, what about RHOMBUS on side A + RHOMBUS on side B? Do the areas of the 2 rhombuses (rhombi?) = the area of the rhombus on side C? Find out. Does anything follow?

Obviously a RECTANGLE as a shape is problematic because the possibilities are unlimited. A key one is the special case of the square, of course, but messing around with other rectangles with proportional relationships of width vs. length is interesting.

Investigate other areas of other shapes and their areas, regular and irregular.

So, how general can we make the theorem?

You can find all their answers, presented via photographs, here.
 
POSTSCRIPT: they weren’t done yet. They not only drew some broad generalizations, they used the figure of yours truly from this blog as a ‘shape’ to explore! Check out all their work here.