7. In $\triangle A B C$, $A B=5, A C=4$, and $\overrightarrow{A B} \cdot \overrightarrow{A C}=12$. Let $P$ be a point on the plane $A B C$. Then the minimum value of $\overrightarrow{P A} \cdot(\overrightarrow{P B}+\overrightarrow{P C})$ is
Rong's matter $\quad-\frac{65}{8}$.
Analysis Given $AB=5, AC=4$, and $\overrightarrow{AB} \cdot \overrightarrow{AC}=12$, we get $\cos A=\frac{3}{5}$.
As shown in the figure, with $A$ as the origin and $AC$ as the $x$-axis, we establish a rectangular coordinate system, then $C(4,0), B(3,4)$, and let $P(x, y)$, so
$$
\begin{aligned}
\overrightarrow{PA} \cdot(\overrightarrow{PB}+\overrightarrow{PC}) & =(-x,-y) \cdot(7-2 x, 4-2 y)=2 x^{2}-7 x+2 y^{2}-4 y \\
& =2\left(x-\frac{7}{4}\right)^{2}+2(y-1)^{2}-\frac{65}{8}
\end{aligned}
$$

Thus, the minimum value of $\overrightarrow{PA} \cdot(\overrightarrow{PB}+\overrightarrow{PC})$ is $-\frac{65}{8}$.