Square in a Pentagon
The diagram at left shows the top of a regular pentagon with the top of a square
inscribed in it. The shapes share a vertex at the top, and the other two
vertices of the square lie on the sides of the pentagon. If the diagram were
continued to include the entire pentagon and the entire square, which shape
would extend below the other?
In
other words, does the whole square fit inside the pentagon, does the square
protrude at the bottom, or do the square and pentagon meet at a single point?
Solution:
The square fits inside the pentagon.
If you are very careful
and draw the square perfectly inside the pentagon, you’ll see that the fourth
vertex of the square lies within the pentagon.
To just see the picture, however, is quite unsatisfying.
After all, if your drawing is off by just a little bit, it could appear that
the fourth vertex lies on the bottom side or even outside the pentagon. How can
you know for sure? You could use trigonometry, but that gets very ugly. Isn’t
there a more elegant approach?
Perhaps the visual “proof” offered by the following series
of folds might convince you.
First, fold the pentagon vertically along the midline.
Next, fold the bottom corners of the pentagon so that the
base lies on the midline. Doing this will form a right angle at the center of
the base.
Why do this? Well, if the fourth
vertex of the square lies exactly on the base of the pentagon, then a right
angle has to touch the midpoint of the base (which is what was just created).
Further, the points at which these two newly formed triangles touch the sides
of the pentagon are the points where the vertices of the square must lie.
Finally, fold the bottom portion of the paper so that a
horizontal fold connects the points where the triangles from the previous
fold meet the sides of the pentagon.
As can be seen, the bottom vertex
does not reach the top of the pentagon. This indicates that a square
inscribed in the pentagon must have a diagonal that is shorter than the
height of the pentagon. Consequently, an inscribed square will lie entirely
within the pentagon.
