Example 2 Let positive numbers $x, y, z$ satisfy the system of equations
$$
\left\{\begin{array}{l}
x^{2}+x y+\frac{y^{2}}{3}=25, \\
\frac{y^{2}}{3}+z^{2}=9, \\
z^{2}+z x+x^{2}=16 .
\end{array}\right.
$$

Find the value of $x y+2 y z+3 z x$.
If we solve the system of three quadratic equations in the conventional way, first finding the values of $x$, $y$, and $z$, and then calculating the value of the algebraic expression, we will inevitably fall into tedious calculations.
In fact, the original system of equations can be transformed into
$$
\left\{\begin{array}{l}
x^{2}+\left(\frac{y}{\sqrt{3}}\right)^{2}-2 x \cdot \frac{y}{\sqrt{3}} \cos 150^{\circ}=5^{2}, \\
\left(\frac{y}{\sqrt{3}}\right)^{2}+z^{2}=3^{2}, \\
z^{2}+x^{2}-2 z x \cos 120^{\circ}=4^{2} .
\end{array}\right.
$$

The above three equations are structurally similar to the cosine rule and Pythagorean theorem, so we can construct
Figure 3, and calculate the areas of
$\triangle A B O$, $\triangle B C O$,
$\triangle C A O$, and $\triangle A B C$,
to find
$$
\begin{array}{l}
x y+2 y z+3 z x \\
=24 \sqrt{3} .
\end{array}
$$