Problem:
A wire is cut into two pieces, one of length and the other of length . The piece of length is bent to form an equilateral triangle, and the piece of length is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is ?
Answer Choices:
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Solution:
Let be the side length of the triangle and the side length of the hexagon. The hexagon can be subdivided into equilateral triangles by drawing segments from the center of the hexagon to each vertex. Because the areas of the large triangle and hexagon are equal, the triangles in the hexagon each have area of the area of the large triangle. Thus
The perimeter of the triangle is and the perimeter of the hexagon is , so
The problems on this page are the property of the MAA's American Mathematics Competitions