Shapes of Four Triangles
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Shapes from Four
Triangles: (Adapted from Shapes and Measurement, written by the
Reconceptualizing Mathematics Project at San Diego State University.)
Given four congruent
isosceles right triangles, how many different polygonal regions can you make,
using all four triangles each time? Two shapes are said to be the same if some
combination of a slide, a rotation, or a flip will transform the first shape
into the second shape. To answer the question of “how many?” you will probably
want to find and display all possible Shapes from Four Triangles. Once you have
done this, your task is to find an argument that proves you have found a complete set of Shapes from Four Triangles.
I.e., how would you explain to someone who isn’t looking at the shapes that i)
your
shapes are all different, and ii)
there
are no more left to be found?
Note: The following three are all the SAME
polygonal region (perhaps called square-with-sail), since some rigid motion
shows they are congruent and hence they are not really different shapes.