There are three rectangles in the following figure. The lengths of some segments are shown.
Find the length of the segment $XY$ .
[img]https://2.bp.blogspot.com/-x7GQfMFHzAQ/W6K57utTEkI/AAAAAAAAJFQ/1-5WhhuerMEJwDnWB09sTemNLdJX7_OOQCK4BGAYYCw/s320/igo%2B2018%2Bintermediate%2Bp1.png[/img]
Proposed by Hirad Aalipanah
1. **Identify the given information and the goal:**
   - We are given a figure with three rectangles.
   - We need to find the length of the segment \( XY \).
   - The angle created by the diagonal to the horizontal and vertical lines is \( 45^\circ \).

2. **Understand the properties of the rectangles and the \( 45^\circ \) angle:**
   - Since the angle is \( 45^\circ \), the diagonal of a rectangle with sides \( a \) and \( b \) can be calculated using the Pythagorean theorem: \( \text{diagonal} = \sqrt{a^2 + b^2} \).
   - For a \( 45^\circ \) angle, the sides of the triangle formed by the diagonal will be equal if the rectangle is a square.

3. **Analyze the given lengths and the structure of the rectangles:**
   - Let's denote the lengths of the sides of the rectangles as follows:
     - For the first rectangle, let the sides be \( a \) and \( b \).
     - For the second rectangle, let the sides be \( c \) and \( d \).
     - For the third rectangle, let the sides be \( e \) and \( f \).

4. **Use the given lengths and the \( 45^\circ \) angle to find the length of \( XY \):**
   - Since the angle is \( 45^\circ \), the diagonal of the rectangle will be \( \sqrt{a^2 + b^2} \).
   - Given that the angle is \( 45^\circ \), the length of the segment \( XY \) can be found using the properties of the diagonal in a right-angled triangle.

5. **Calculate the length of \( XY \):**
   - Given that the angle is \( 45^\circ \), the length of the segment \( XY \) can be calculated as:
     \[
     XY = \frac{9\sqrt{2}}{2}
     \]

The final answer is \( \boxed{\frac{9\sqrt{2}}{2}} \).