10.046. A circle touches the larger leg of a right triangle, passes through the vertex of the opposite acute angle, and has its center on the hypotenuse of the triangle. What is the radius of the circle if the lengths of the legs are 5 and $12?$
Solution.

Given $A C=5, C B=12$ (Fig. 10.46). In $\triangle A B C$, $\angle C=90^{\circ}$, $A B=\sqrt{25+144}=13$. We have $\triangle A C B \sim \triangle O D B \Rightarrow \frac{A B}{O B}=\frac{A C}{O D}$. Let $O D=R$ be the radius of the given circle, then $\frac{13}{O B}=\frac{5}{R}$ and $O B=13-R$, so $\frac{13}{13-R}=\frac{5}{R} ; R=\frac{65}{18}$.

Answer: $\frac{65}{18}$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-578.jpg?height=464&width=306&top_left_y=92&top_left_x=301)

Fig. 10.47

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-578.jpg?height=470&width=306&top_left_y=77&top_left_x=767)

Fig. 10.48