## Task A-1.2.

Given is a right triangle $A B C$ with a right angle at vertex $C$ and side lengths $|A B|=26$, $|B C|=24$. A semicircle is inscribed in it with the diameter on side $\overline{B C}$ which contains point $C$. The semicircle touches side $\overline{A B}$. What is the radius of the inscribed semicircle?
Scoring: 1 point is awarded for $|A C|=10$ and 1 point for the correct result. The main idea of the solution is worth 4 points.

In the first solution, 1 point is awarded for the conclusion $|A D|=|A C|$, 1 point for the justification of this conclusion, 1 point for applying the Pythagorean theorem to triangle $B D S$, and 1 point for using that $|C S|=r$ and obtaining an equation in which the only unknown is $r$.

In the second solution, 2 points are awarded for recognizing similarity, 1 point for writing the ratio that arises from this similarity, and 1 point for using that $|C S|=r$ and obtaining an equation in which the unknown is $r$.

In the third solution, 1 point is awarded for expressing the area of triangle $A B C$ as the sum of the areas of triangles $A S B$ and $A S C$, 1 point for using that $|C S|=|D S|=r$ in the area formula, and 2 points for obtaining an equation with $r$ as the unknown.