2. (7 points) In rectangle $A B C D$, side $A B$ is equal to 6, and side $B C$ is equal to 11. Bisectors of the angles from vertices $B$ and $C$ intersect side $A D$ at points $X$ and $Y$ respectively. Find the length of segment $X Y$.

![](https://cdn.mathpix.com/cropped/2024_05_06_2509cbb2376df96befe7g-1.jpg?height=317&width=417&top_left_y=1121&top_left_x=822)

## Answer: 1.
Solution. Angles $A X B$ and $X B C$ are equal as alternate interior angles when lines $A D$ and $B C$ are parallel and line $B X$ is the transversal. Angles $X B C$ and $X B A$ are equal since $B X$ is the bisector of angle $A B C$. We obtain that $\angle A X B = \angle X B A$, which means triangle $A X B$ is isosceles, $A B = A X = 6$; $X D = A D - A X = 11 - 6 = 5$. Similarly, we get that $A Y = 5$. Then $X Y = A D - A Y - X D = 11 - 5 - 5 = 1$.

Criteria. Any correct solution: 7 points.

Proved that $A Y = 5$, but the length of segment $X Y$ is not found or found incorrectly: 4 points.

Proved that triangle $A B X$ is isosceles and no further progress: 2 points.

Only the correct answer is provided: 1 point.