Question 1112136
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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.<br>
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.<br>
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.<br>
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.<br>
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.<br>
Answer:  The ratio of the area of hexagon C to the area of hexagon A is 9:16.