5. (10 points) As shown in the figure, in the right triangle $A B C$, $\angle A B C=90^{\circ}, A B P A^{\prime} B^{\prime}, B C P B^{\prime} C^{\prime}, A C P A^{\prime}$ $C^{\prime}$, and the distances between the three pairs of parallel lines are all 1. If $A C=10, A B=8, B C=6$, find the maximum distance from a point on triangle $A^{\prime} B^{\prime} C^{\prime}$ to the three sides of triangle $A B C$.
【Analysis】Let the point be $P$. If point $P$ is on $A^{\prime} C^{\prime}$, and the distances from $P$ to sides $A B$ and $B C$ are $a$ and $b$ respectively, then the sum of the distances from $P$ to the three sides of triangle $A B C$ is $a+b+1$. Connecting $A P$, $B P$, and $C P$, since the areas of triangles $P A B$, $P B C$, and $P A C$ sum to 24, we have $5+3 b+4 a=24$, which gives $b=\frac{19-4 a}{3}$, and $a+b+1=\frac{22-a}{3}$. We then discuss different cases.

【Solution】Let the point be $P$. If point $P$ is on $A^{\prime} C^{\prime}$, and the distances from $P$ to sides $A B$ and $B C$ are $a$ and $b$ respectively, then the sum of the distances from $P$ to the three sides of triangle $A B C$ is $a+b+1$ (1). Connecting $A P$, $B P$, and $C P$, since the areas of triangles $P A B$, $P B C$, and $P A C$ sum to 24, we have $5+3 b+4 a=24$, which gives $b=\frac{19-4 a}{3}$, and $a+b+1=\frac{22-a}{3}$. We then discuss different cases.

When $a=1$, $a+b+1=7$, achieving the maximum value of 7;
If point $P$ is on $B^{\prime} C^{\prime}$, using the same method, we get $a+c+1=\frac{a+26}{5}<\frac{6+26}{5}<7$.
If point $P$ is on $A^{\prime} B^{\prime}$, using the same method, we get $b+c+1=\frac{2}{5} b+5$, and it is easy to see that the maximum value of $b$ is 5, so the maximum value of the distance sum in this case is also 7.

In summary, the maximum value of the sum of the distances from a point on triangle $A^{\prime} B^{\prime} C^{\prime}$ to the three sides of triangle $A B C$ is 7;
Answer: The maximum value of the sum of the distances from a point on triangle $A^{\prime} B^{\prime} C^{\prime}$ to the three sides of triangle $A B C$ is 7.