Track / Stage 3 / 69 of 260 #549 of 2444

Problem 549

AMC 10/12, early questions

Geometry Difficulty 3.4 Multiple choice Slovenia — National Math Olympiad · Slovenia

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

Pick one

Official solution

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

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.