Problem:
Each of the sides of a square with area is bisected, and a smaller square is constructed using the bisection points as vertices. The same process is carried out on to construct an even smaller square . What is the area of ?
Answer Choices:
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Solution:
The sides of have length , so by the Pythagorean Theorem the sides of have length . By similar reasoning the sides of have length . Thus the area of is .
Connect the midpoints of the opposite sides of . This cuts into congruent squares as shown. Each side of cuts one of these squares into two congruent triangles, one inside and one outside.
Thus the area of is half that of . By similar reasoning, the area of is half that of , and one fourth that of .
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The problems on this page are the property of the MAA's American Mathematics Competitions