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Italian Mathematical Olympiad, 2003

Geometry Difficulty 3.6 AMC 10/12 Find the answer Italy

Problem:

What is the measure of the radius of the small circle in the figure alongside, if the side of the square is $1$ long?

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Solution

Solution:

The answer is (A). With reference to the figure alongside, letting $r$ be the required radius, we have $1 + r + r \sqrt{2} = \sqrt{2}$ , from which $r = \frac{\sqrt{2} - 1}{\sqrt{2} + 1}$ and finally, rationalizing, $r = 3 - 2 \sqrt{2}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.