4. Answer. 6 cm. Solution. Let $A B=9, B C=4, A C=x, S$ - the area of triangle $A B C$. Then the heights of the triangle are $2 S / 4, 2 S / 9, 2 S / x$. By the triangle inequality, side $A C$ can be either the largest or the middle in length. Applying the triangle inequality to the triangle formed by the heights, we see that among the quantities $2 S / 4, 2 S / 9, 2 S / x$, the quantity $2 S / x$ can also be either the largest or the middle. However, the larger the side, the smaller the height drawn to it, and the height to the largest side can only be the smallest. Therefore, the side $x$ (and the height drawn to it) can only be the middle. This allows us to write the proportion $9: x = 2 S / 4: 2 S / x$ and $x=6$. Grading criteria. Full solution - 7 points. The proportion is written correctly, but not justified (in any way) - 4 points. In all other cases (including an answer without a solution) - 0 points.
4. Triangle $A B C$ is similar to the triangle formed by its altitudes. Two sides of triangle $A B C$ are 4 cm and 9 cm. Find the third side.