4.1. (12 points) In rectangle $A B C D$, point $E$ is marked on the extension of side $C D$ beyond point $D$. The bisector of angle $A B C$ intersects side $A D$ at point $K$, and the bisector of angle $A D E$ intersects the extension of side $A B$ at point $M$. Find $B C$, if $M K=8$ and $A B=3$.
Answer: $\sqrt{55} \approx 7.42$.

Solution. Since $B K$ is the bisector of angle $A B C$, triangle $A B K$ is isosceles, $A B = A K$. Similarly, we get that $A D = A M$. Therefore, triangles $A K M$ and $A B D$ are equal by two legs. Applying the Pythagorean theorem in triangle $A K M$, we find $B C = A D = A M = \sqrt{M K^{2} - A B^{2}} = \sqrt{55}$.