jmc

Question

A hexagon is drawn with its vertices at (0,0),(1,0),(2,1),(2,2),(1,2), and (0,1), and all of its diagonals are also drawn, as shown below. The diagonals cut the hexagon into 24 regions of various shapes and sizes. These 24 regions are shown in pink and yellow below. If the smallest region (by area) has area a, and the largest has area b, then what is the ratio a:b? Give your answer in lowest terms. FIGURE

Accepted Answer

We add three lines to the diagram, connecting midpoints of opposite edges of the hexagon: FIGURE We have also shaded a triangle above. The shaded triangle is now divided into six regions of equal area by its medians. In similar fashion, the whole hexagon is divided into 36 regions of equal area. Each of the original 24 regions covered one or two of these 36 new regions, so the ratio of the smallest to the largest area among the original 24 regions is 1:2.