Question 1112136: A regular hexagon A has the midpoints of its edges 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? Please Explain.
Answer by greenestamps(13200)   (Show Source):
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The ratio of the area of hexagon A to the area of hexagon B is going to be the same as the ratio of the area of hexagon B to the area of hexagon C, because the same process is used to create each smaller hexagon.
For hexagon A and hexagon B, choose one side of hexagon A and draw segments from the center of the hexagon to the midpoint of that side and to one endpoint of that side.
Those two segments, along with the half side of hexagon A, form a 30-60-90 right triangle, so the ratio of the lengths of the two segments is sqrt(3):2.
But those two segments are corresponding parts of hexagons A and B. Since the ratio of corresponding linear measurements between hexagons B and A is sqrt(3):2, the ratio of the areas of hexagons B and A is the square of that ratio, which is 3:4.
So the area of hexagon B is 3/4 the area of hexagon A; and similarly the area of hexagon C is 3/4 the area of hexagon B. So the area of hexagon C is (3/4)^2 = 9/16 the area of hexagon A.
Answer: The ratio of the area of hexagon C to the area of hexagon A is 9:16.
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