10.069. The radius of the circle circumscribed around a right-angled triangle is $15 \mathrm{cm}$, and the radius of the circle inscribed in it is 6 cm. Find the sides of the triangle.
## Solution.

By the condition $A B=2 R=2 \cdot 15=30$ (cm) (Fig. 10.68). The radius of the inscribed circle $r=\frac{a+b-c}{2}$. Hence, $2 r=a+b-c$. Solving the system

$$
\left\{\begin{array}{l}
2 r = a + b - c, \\
a^{2} + b^{2} = c^{2}
\end{array} \Leftrightarrow \left\{\begin{array}{l}
12 = a + b - 30 \\
a^{2} + b^{2} = 900
\end{array}\right.\right.
$$

we get $a_{1}=18$ (cm), $b_{1}=24$ (cm) or $a_{2}=24$ (cm), $b_{2}=18$ (cm).

Answer: 18; 24 and 30 cm.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-589.jpg?height=390&width=324&top_left_y=93&top_left_x=176)

Fig. 10.69

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-589.jpg?height=396&width=450&top_left_y=102&top_left_x=717)

Fig. 10.70