2003 AMC 12B/Problem 8
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Equilateral is inscribed in equilateral with perpendicular to . What is the ratio of the area of to the area of ?
With some quick angle chasing, we can find that it is also true that is perpendicular to and is perpendicular to . Then we have and three congruent (by AAS congruency) triangles making up . So, letting , which is permissible since we only want the ratio of the areas, and all equilateral triangles are similar, we have that and . So we want:
It follows that the answer is .