Problem:
Let be a square. Let , and be the centers, respectively, of equilateral triangles with bases , and , each exterior to the square. What is the ratio of the area of square to the area of square
Answer Choices:
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Solution:
Without loss of generality, let the square and equilateral triangles have side length 6 . Then the height of the equilateral triangles is , and the distance of each of the triangle centers, , and , to the square is . It follows that the diagonal of square has length , and the diagonal of square has length equal to the side length of square plus twice the distance from the center of an equilateral triangle to square or . The required ratio of the areas of the two squares is equal to the square of the ratio of the lengths of the diagonals of the two squares, or
OR
Without loss of generality, place the square in the Cartesian plane with coordinates , and . The center of each equilateral triangle is the point at which the medians intersect, and this point is one third of the way from the midpoint of a side of the triangle to the opposite vertex. The height of an equilateral triangle with side 6 is , so the centers are units from the sides of the square. Therefore the coordinates are , , and . The area of square is half the product of the lengths of its diagonals, or . Square has area , so the desired ratio is .
The problems on this page are the property of the MAA's American Mathematics Competitions