On the extension of side $A B$ of rhombus $A B C D$ beyond point $B$, a point $M$ is taken such that $M D=M C$ and $\angle M D C=\operatorname{arctg} 8 / 5$. Find the ratio of segments $M A$ and $M B$.
Let the side of the rhombus be denoted as $10 x$. Let $P$ be the projection of point $M$ on $D C$, and $K$ be the projection of point $D$ on $A B$. Since triangle $D M C$ is isosceles, $D P = 5 x$. Then, $M P = D P \operatorname{tg} \angle M D C = 8 x, D K = M P = 8 x, A K^{2} = A D^{2} - K D^{2} = 36 x^{2}, M K = D P = 5 x, A M = A K + M K = 11 x, M B = A M - A B = x, A M: M B = 11: 1$.

![](https://cdn.mathpix.com/cropped/2024_05_06_633abcb7b0e2bd041761g-03.jpg?height=358&width=784&top_left_y=1233&top_left_x=640)

## Answer

$11: 1$.