10.016. In the sector $A O B$ with radius $R$ and angle $90^{\circ}$, a circle is inscribed, touching segments $O A, O B$ and arc $A B$ (Fig. 10.16). Find the radius of the circle.
## Solution.

$\triangle O M F$ is isosceles $\left(\angle M O B=\angle M F O=45^{\circ}\right)$. Therefore, $M O=M F$, $O F=R \sqrt{2}$. From $\triangle O_{1} Q O \sim \Delta O M F$ we have $\frac{r}{R}=\frac{R-r}{\sqrt{2} R}, r=R(\sqrt{2}-1)$.

Answer: $\quad R(\sqrt{2}-1)$

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-564.jpg?height=356&width=370&top_left_y=142&top_left_x=95)

Fig. 10.17
Fig. 10.18

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-564.jpg?height=396&width=438&top_left_y=196&top_left_x=445)

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-564.jpg?height=430&width=388&top_left_y=93&top_left_x=860)

Fig. 10.19