4. Given point $A(-3,4)$, moving point $B$ on the $y$-axis, moving point $C$ on the line $l: x+y=0$, then the minimum value of the perimeter of $\triangle A B C$ is
A. $5 \sqrt{2}$
B. $4+\sqrt{10}$
C. $3+2 \sqrt{2}$
D. 7
Answer
As shown in the figure, construct the symmetric points $M, N$ of point $A$ with respect to the $y$-axis and the line $y=x$, respectively. Then the length of line segment $M N$ equals the perimeter of $\triangle A B C$. It is easy to know that
$$
M(3,4), N(-4,3) \Rightarrow|M N|=\sqrt{7^{2}+1^{2}}=5 \sqrt{2} \text {, }
$$

Therefore, the answer is $A$.