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2008 AMC 10A Problem 10

Problem 10 of 25EasierGeometry

Each of the sides of a square S1S_1 with area 1616 is bisected, and a smaller square S2S_2 is constructed using the bisection points as vertices. The same process is carried out on S2S_2 to construct an even smaller square S3.S_3. What is the area of S3?S_3?

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Solution

The side of S1S_1 is 4.4. By the Pythagorean theorem, the side of S2S_2 is 22+22=22,\sqrt{2^2 + 2^2} = 2\sqrt{2}, so its area is 8.8. By the same reasoning, S3S_3 has half the area of S2,S_2, namely 4.4. Thus, the correct answer is E.

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Concepts: square (geometry) · Pythagorean Theorem · area ratio

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.