# 2016 AMC 12A Problems/Problem 17

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## 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 ,