[ Properties and characteristics of an isosceles triangle. ]

Through the vertices $A$ and $C$ of triangle $A B C$, lines are drawn perpendicular to the bisector of angle $A B C$ and intersect the lines $C B$ and $B A$ at points $K$ and $M$ respectively. Find $A B$, if $B M=8, K C=1$.
Let point $K$ be located on side $B C$ of triangle $A B C$. Triangles $A B K$ and $M B C$ are isosceles (the bisector drawn from vertex $B$ is also the height), so $A B=B K=B C-C K=8-1=7$.

If point $K$ is located on the extension of segment $B C$ beyond point $C$, then similarly we find that $A B=9$.

## Answer

7 or 9.