Right triangle $A B C$ is divided by the height $C D$, drawn to the hypotenuse, into two triangles: $B C D$ and $A C D$. The radii of the circles inscribed in these triangles are 4 and 3, respectively. Find the radius of the circle inscribed in triangle $A B C$.
Let $r$ be the desired radius, $r_{1}$ and $r_{2}$ be the radii of the given circles. From the similarity of triangles $A C D$, $C B D$, and $A B C$, it follows that $r_{1}: r_{2}: r = A C: B C: A B$. Therefore, $r^{2} = r_{1}^{2} + r_{2}^{2} = 25$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-36.jpg?height=346&width=552&top_left_y=1348&top_left_x=753)

## Answer

5. 

Send a comment