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Geometry Difficulty 3.4 AMC 10/12 Find the answer Slovenia

Three equal circles are inscribed into the equilateral triangle with the sides of length 11 (see figure). What is the length of the radii of the circles?
Figure 1

Pick one

Solution

If we denote the radii by rr, we have r3+2r+r3=1r\sqrt{3} + 2r + r\sqrt{3} = 1. It follows that r=12(1+3)=3−14r = \frac{1}{2(1+\sqrt{3})} = \frac{\sqrt{3}-1}{4}.

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