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?

Algebra ->  Finance -> 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?      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
.

https://artofproblemsolving.com/wiki/index.php/1996_AHSME_Problems/Problem_19



Answer by greenestamps(13200) About Me  (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 sqrt%283%29%3A2; so the ratio of the areas of the two hexagons is 3:4.

ANSWER: 3:4