5. The angle between the lateral sides $A B$ and $C D$ of trapezoid $A B C D$ is $30^{\circ}$. A circle is inscribed in the trapezoid, and the side $A B$ is divided by the point of tangency into segments of length $\sqrt{3}$ and $3 \sqrt{3}$. Determine the length of the lateral side $C D$ of the trapezoid.
Answer: 6 or 12.

Solution: A similar solution to this problem is present in Variant 1 under the same number.

## Evaluation criteria are provided in the table:

| Points | Evaluation Criteria |
| :---: | :--- |
| **7** | Fully justified solution. |
| **6** | Justified solution with minor flaws. |
| **5 - 6** | The solution contains minor errors, gaps in justification, but is generally correct and can become fully correct after minor corrections or additions. |
| **4** | The problem is more solved than not, for example, one of two (more |
|  | complex) significant cases is correctly considered. |
| **2-3** | The problem is not solved, but formulas, diagrams, considerations, or auxiliary statements relevant to the solution of the problem are provided. |
| **1** | The problem is not solved, but an attempt at a solution has been made, for example, individual (partial) cases have been considered in the absence of a solution or with an incorrect solution. |
| **0** | The solution is absent, or the solution does not meet any of the criteria listed above. |

Drawings and graphs for the problems.

8th grade. Problem 4.

![](https://cdn.mathpix.com/cropped/2024_05_06_1d90a55096de80c9e35eg-7.jpg?height=622&width=1310&top_left_y=503&top_left_x=430)

Fig. 1

9th grade. Problem 5.

![](https://cdn.mathpix.com/cropped/2024_05_06_1d90a55096de80c9e35eg-7.jpg?height=557&width=548&top_left_y=1523&top_left_x=503)

Fig. 2

![](https://cdn.mathpix.com/cropped/2024_05_06_1d90a55096de80c9e35eg-7.jpg?height=565&width=511&top_left_y=1505&top_left_x=1229)

Fig. 3