Four larger circles with radius 5 are arranged so that their centres are the vertices of a square. Each of the larger circles is tangent to (that is, just touches) two of the other circles, as shown. A smaller circle with radius $r$ is drawn in the region between the four larger circles. The smaller circle is tangent to each of the larger circles. The value of $r$ is closest to
(A) 1.9
(B) 2.0
(C) 2.1
(D) 2.2
(E) 2.3

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-004.jpg?height=366&width=374&top_left_y=1226&top_left_x=1301)

Part C: Each correct answer is worth 8.

Each correct answer is an integer from 0 to 99, inclusive.

A one-digit answer (such as " 7 ") must be coded with a leading zero (" 07 ").

Note: The integer formed by the rightmost two digits of 12345 is 45 . The integer formed by the rightmost two digits of 6307 is 7 , coded 07 .
Draw one of the diagonals of the square. The diagonal passes through the centre of the square.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-005.jpg?height=363&width=371&top_left_y=922&top_left_x=931)

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

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

ANSWER: $(\mathrm{C})$