, 2023

Draw one of the diagonals of the square. The diagonal passes
through the centre of the square.

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By symmetry, the centre of the smaller circle is the centre of the
square. (If it were not the centre of the square, then one of the four
larger circles would have to be different from the others somehow, which
is not true.)

Further, the diagonals of the square pass through the points where the
smaller circle is tangent to the larger circles. (The line segment from
each vertex of the square to the centre of the smaller circle passes
through the point of tangency. These four segments are equal in length
and meet at right angles since the diagram can be rotated by 90 degrees
without changing its appearance. Thus, each of these is half of a
diagonal.)

Since each of the larger circles has radius 5, the side length of the
square is $5 + 5 = 10$.

Since the square has side length 10, its diagonal has length $\sqrt{10^2 + 10^2} = \sqrt{200}$ by the
Pythagorean Theorem.

Therefore, $5 + 2r + 5 = \sqrt{200}$
which gives $2r = \sqrt{200} - 10$
and so $r \approx 2.07$.

Of the given choices, $r$ is closest
to 2.1, or (C).