The given question seems to involve a right-angled triangle $ABC$, where $A \hat B C = 90^\circ$. The triangle includes the medians $AM$ and $BN$, with $M$ and $N$ being the midpoints of sides $BC$ and $AC$, respectively. The goal is to determine the ratio of the length of $BN$ to $AM$, with the given value of $AB = 2\sqrt{3}$.

Let me interpret and solve the problem:

1. Understanding the given triangle: Since $A \hat B C = 90^\circ$, triangle $ABC$ is a right triangle. We know:
   • (