Guarding the Ring

Geometry Level 3

An art museum wants to build a new gallery. The floor plan is an annulus (a ring shape bounded by two concentric circles), with walls constructed on both circles. The outer circle has a radius of \(10\text{ m}\), and we are not able to see through the inner circle.

The museum will hire guards to ensure that the public does not touch the artwork. Each guard will stand at a fixed spot, but can look in all directions from that spot.

If they can afford to hire only 4 guards, and every part of the gallery must be visible to at least one guard, then what is the maximum (supremum) possible radius of the inner circle in meters?

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