# A Ball Above the Planes

**Geometry**Level 5

\[ \begin{eqnarray} && x + 3 y + 2 z = 5 \\ && 2x - y + z = 5 \\ && x + y - z = 5\end{eqnarray} \]

In the following, it is assumed the that the \( xy \) plane lies horizontally, and that the \( z \) axis points upward.

Three planes are given as shown above.

A ball of radius 10 units is placed above the three planes such that it is tangent to all three of them.

What is the \(z \) coordinate of the top of the ball?