Difference between revisions of "2016 AMC 12A Problems/Problem 17"
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Revision as of 13:33, 4 February 2016
Problem 17
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
?
Solution
The center of an equilateral triangle is its centroid, where the three medians meet.
The distance along the median from the centroid to the base is one third the length of the median.
Let the side length of the square be . The height of
is
so the distance from
to the midpoint of
is
(from above)
(side length of the square).
Since is the diagonal of square
,