10. Define the "coordinate distance" between two points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ as:
$$
d(P, Q)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right| .
$$

If $C(x, y)$ has equal coordinate distances to points $A(1,3)$ and $B(6,9)$, where the real numbers $x$ and $y$ satisfy $0 \leqslant x \leqslant 10$, $0 \leqslant y \leqslant 10$, then the sum of the lengths of all possible trajectories of point $C$ is $\qquad$
10.5 $(\sqrt{2}+1)$.

From the condition, we get
$$
|x-1|+|y-3|=|x-6|+|y-9| \text {. }
$$

When $x \leqslant 1, y \geqslant 9$, there is no solution;
When $1 \leqslant x \leqslant 6, y \geqslant 9$, there is no solution;
When $x \geqslant 6, y \geqslant 9$, there is no solution;
When $x \leqslant 1,3 \leqslant y \leqslant 9$, $y=8.5$, the length of the line segment is 1;
When $1 \leqslant x \leqslant 6,3 \leqslant y \leqslant 9$, $x+y=9.5$, the length of the line segment is $5 \sqrt{2}$;
When $x \geqslant 6,3 \leqslant y \leqslant 9$, $y=3.5$, the length of the line segment is 4;
When $x \leqslant 1, y \leqslant 3$, there is no solution;
When $1 \leqslant x \leqslant 6, y \leqslant 3$, there is no solution;
When $x \geqslant 6, y \leqslant 3$, there is no solution.
In summary, the sum of the lengths of the line segments formed by point $C$'s trajectory is $1+5 \sqrt{2}+4=5(\sqrt{2}+1)$.