Example 4 Algebraic expression
$$
\sqrt{9 x^{2}+4}+\sqrt{9 x^{2}-12 x y+4 y^{2}+1}+\sqrt{4 y^{2}-16 y+20}
$$

reaches its minimum value when $x$ and $y$ are $\qquad$ .
(2005, I Love Mathematics Junior High School Summer Camp Mathematics Competition)
Solve as shown in Figure 1.
Original expression
\[
\begin{aligned}
= & \sqrt{\left[0-(-2)^{2}\right]^{2}+(3 x-0)^{2}}+ \\
& \sqrt{(1-0)^{2}+(2 y-3 x)^{2}}+ \\
& \sqrt{(3-1)^{2}+(4-2 y)^{2}} \\
= & A B+B C+C D \geqslant A D,
\end{aligned}
\]

where, \( A(-2,0), B(0,3 x), C(1,2 y), D(3,4) \), and when points \( B \) and \( C \) are on the line segment \( A D \), the original expression achieves its minimum value, at this time,
\[
\frac{3 x}{2}=\frac{4}{5}, \frac{2 y}{3}=\frac{4}{5}.
\]

Solving for \( x \) and \( y \) yields \( x=\frac{8}{15}, y=\frac{6}{5} \).