1996 AHSME Problems/Problem 19
The midpoints of the sides of a regular hexagon are joined to form a smaller hexagon. What fraction of the area of is enclosed by the smaller hexagon?
is copied six times to form the hexagon, so if we find the ratio of the area of the kite inside to the the area of itself, it will be the same ratio.
Let so that the area of the triangle is .
Notice that is made up of a kite and two triangles. The two hypotenuses of these two triangles form , so the hypotenuse of each triangle must be . Thus, the legs of each triangle are and , and the area of two of these triangles is .
Subtracting the area of the two triangles from the area of the equilateral triangle, we find that the area of the kite is .
Thus, the ratio of areas is , which is option .
We see six isosceles triangles at the vertices of the large hexagon. If we let the side of the larger hexagon be , we find that the two congruent sides of these triangles are both . By the Law of Sines, we find that the third side is:
This third side of the triangle is the length of the edge of the hexagon. The ratio of the sides of the small hexagon to the large hexagon is , since the large hexagon has a unit side. The ratio of the areas is the square of the ratio of the sides, which is , or option .
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