2. Find the area of a right triangle if the height drawn from the right angle divides it into two triangles, the radii of the inscribed circles of which are 3 and 4.
Answer: 150.

Solution. The ratio of the radii of the inscribed circles is equal to the similarity coefficient of the right triangles into which the height divides the original triangle. This coefficient is equal to the ratio of the legs of the original triangle. Let these legs be denoted as $3x$ and $4x$. By the Pythagorean theorem, the hypotenuse of the original triangle is $5x$, and the radius of the inscribed circle, due to similarity, is 5. Using the equality $2r = a + b - c$ for the radius of the inscribed circle (where $a$ and $b$ are the legs, and $c$ is the hypotenuse), we get the equation: $10 = 3x + 4x - 5x$, from which $x = 5$. Thus, the legs of the original triangle are 15 and 20, and its area is $S = \frac{1}{2} \cdot 15 \cdot 20 = 150$.