### Overlapping polygons - January 1997

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In December there was an interesting problem used for the Geometry Problem of the Week:

Two congruent 10cm x 10cm squares overlap. A vertex of one square is at the center of the other square. What is the largest possible value for the area where they overlap? (The one square is movable, as long as the vertex remains in the center.)
The answer is that the overlap will always be 25 cm^2, and there are a number of cool ways to explain it.

A couple of people wondered whether the same thing would work for other polygons like triangles, rectangles, pentagons, etc. Would the figures have to be regular polygons? Would it work for a rhombus, for example? I thought that was a good question, so that's what I'm asking this month.

First, figure out a way to explain the squares problem above so that anybody would understand the answer - anyone would take one look at your argument and be totally convinced.

Now explore the same problem for other polygons. Make sure you look at some regular and irregular ones. See what you can learn about which polygons will have answers similar to the squares answer, and explain as clearly as you can.

- Annie Fetter