1. Sketch a figure that is gnomon to a rectangle of length 2 and width 1; label any sides with appropriate lengths.

   Solution:
   Begin by sketching the original rectangle:
   

   
   
   • We could draw an L-shaped gnomon on two sides of our original shape. There are an infinite number of ways to do this -- the inner part of the L has to be 2 long and 1 wide in order to fit around our original rectangle, but as long as the resulting final rectangle is twice as long as it is wide, any such L will do. Below is one where the final rectangle is similar to the original with a scale factor of 1.5:
     
     
   • Or we can attach a rectangle to the long side of the original. If we do it this way, then we don't have any choices about dimensions.
     
     Since the side of length 2 now has to be the shorter side, and since the final result has to be similar to the original, the ratio of the new long side to the original long side (new/2) must equal the ratio of the new short side to the original short side (2/1) and so the new side must have length 4. Thus the attachment must have length 3.

   • Can we follow the same idea but attach the gnomon to the short side of our original rectangle?
     When we attach the gnomon to the short side, the newly formed rectangle has a longer long side (2+?) but the same length short side (1) as the original. There is no (non-zero) value for ? that will make new/2=1/1=1.

     It is impossible to add a gnomon to the short side of a rectangle.

2. Sketch a figure that is gnomon to a circle of radius 3. Label any relevant portions with appropriate lengths.

   Solution: Since all circles are similar, anything that we can attach to original circle to form a new circle will be a gnomon to the original circle.

   
3. Do any rectangles have square gnomons? If so, which? That is, what can you say about a rectangle that has a square gnomon?
   • We already know that not all rectangles have square gnomons: If we want to attach a square gnomon to a rectangle and create a new rectangle that's similar to the first, we have to attach it to one of the sides.

     But when we tried attaching a gnomon to our 2 x 1 rectangle, we discovered that we couldn't attach a gnomon to the short side, and that if we tried attaching it to the long side, we had no choice over the size of the gnomon -- it had to be a 3 x 2 gnomon.

             
   • We also already know (from the above) that in general, we can't attach a rectangular or square gnomon to the short side of a rectangle -- there's no way the newly formed rectangle could be similar to the original.

     So in order for a rectangle to have a square gnomon, the square gnomon would have to attach to the long side of the original rectangle.

     

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