10.2. On a plane, a semicircle with diameter $A B=36$ cm was constructed; inside it, a semicircle with diameter $O B=18$ cm was constructed ($O-$ the center of the larger semicircle). Then, a circle was constructed that touches both semicircles and the segment АО. Find the radius of this circle. Justify your answer.
Solution: Let the desired radius be $x$. Let points $C$ and $Q$ be the centers of the smaller semicircle and the inscribed circle, respectively, and $D$ be the point of tangency of the inscribed circle with the diameter $AB$. Due to the tangency of the circle and the larger semicircle, the ray $OQ$ passes through the point of tangency, so $OQ = 18 - x$. Due to the tangency of the circle and the smaller semicircle, the segment $CQ$ passes through the point of tangency, so $CQ = 9 + x$. Due to the tangency of the circle and the segment $AB$, the segment $QD$ equals $x$ and is the height of triangle $QOC$. In triangle $QOC$, all three sides and the height are expressed through $x$. We will find the area of the triangle in two ways. The semiperimeter of the triangle is $\frac{9 + (18 - x) + (9 + x)}{2} = 18$ and by Heron's formula, its area is $\sqrt{18 \cdot (18 - 9) \cdot (18 - (18 - x)) \cdot (18 - (9 + x))} = \sqrt{18 \cdot 9 \cdot x \cdot (9 - x)}$. This area is also equal to half the product of the height and the side to which the height is drawn, that is, it is equal to $\frac{9 \cdot x}{2}$. From the equation $\sqrt{18 \cdot 9 \cdot x \cdot (9 - x)} = \frac{9 \cdot x}{2}$, we find $x = 8$.

Answer: 8 cm

Recommendations for checking:

| present in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| The correct approach to the solution resulted in an incorrect answer due to errors in calculations (algebraic or arithmetic) | 6 points |
| The elements of triangle $QOC$ are correctly written (in terms of the unknown), fully defining the triangle, but the solution is not completed to the answer | 5 points |
| Triangle $QOC$ is considered, but not all elements defining it are found (expressed) | 3 points |
| The correct application of the coordinate method, which is not completed to the answer (depending on the degree of progress) | 1 - 3 points |
| Correct answer without justification and/or any calculations from which the solution process is not clear | 0 points |