SOLUTION: In a regular hexagon, the midpoints of the sides are joined to form a shaded regular hexagon. What fraction of the larger hexagon is shaded?

Question 1182756: In a regular hexagon, the midpoints of the sides are joined to form a shaded regular hexagon. What fraction of the larger hexagon is shaded?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
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https://artofproblemsolving.com/wiki/index.php/1996_AHSME_Problems/Problem_19

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

I think both the solutions in the link provided by the other tutor are more complicated than necessary.

Let M be the midpoint of side AB of the original regular hexagon. Then M is one vertex of the smaller inscribed regular hexagon.

Let O be the center of the two hexagons; draw segments OA and OM; triangle AMO is a 30-60-90 right triangle.

OA and OM are corresponding parts of the two hexagons; the ratio of their lengths is ; so the ratio of the areas of the two hexagons is 3:4.

ANSWER: 3:4

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