document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #727232 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "Answer: The ratio of the area of hexagon C to the area of hexagon A is 9:16. \n" ); document.write( "

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