Question 1180867: A regular hexagon A has the midpoints of its edge joined to form a smaller hexagon B, and this process is repeated by joining the midpoints of the edges of B to get a third hexagon C. What is the ratio of the area of C to the area of A
Answer by greenestamps(13200)   (Show Source):
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Make a sketch and draw segments from the centers of the hexagons to each vertex of both hexagons.
Neighboring segments, along with the sides of hexagon A, form 30-60-90 degree triangles. So in those triangles, the length of a segment to the vertex of hexagon A and the length of a segment to the vertex of hexagon B are in the ratio 2:sqrt(3) (hypotenuse and long leg of a 30-60-90 right triangle).
Those segments are corresponding parts of hexagon A and hexagon B; the ratio of the areas of the two hexagons is the square of the ratio of the lengths of the corresponding sides.
So the ratio of the areas of hexagons A and B is 4:3.
Of course you will have the same ratio of areas when you construct hexagon C using the midpoints of the sides of hexagon B.
So when hexagons are inscribed in hexagons as described in this problem, the area of each hexagon is 3/4 of the area of the previous hexagon.
So the ratio of the area of hexagon C to the area of hexagon A is (3/4)^2 = 9/16, or 9:16.
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