Problem:
A square and an equilateral triangle have the same perimeter. Let be the area of the circle circumscribed about the square and be the area of the circle circumscribed about the triangle. Find .
Answer Choices:
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Solution:
Rescaling to different units does not affect the ratio of the areas, so let the perimeter be 12. Each side of the square then has length 3 , and each side of the triangle has length 4 . The diameter of the circle circumscribing the square is the diagonal of the square, . Thus . The altitude of the triangle is , so the radius of the circle circumscribing the triangle is , and . Therefore
The problems on this page are the property of the MAA's American Mathematics Competitions