10.283. The area of a right-angled triangle is $24 \mathrm{~cm}^{2}$, and the hypotenuse is 10 cm. Find the radius of the inscribed circle.
Solution.

Let $a$ and $b$ be the legs of the triangle. We have the system $\left\{\begin{array}{l}a b=48, \\ a^{2}+b^{2}=100,\end{array} \Rightarrow\right.$ $\Rightarrow a^{2}+2 a b+b^{2}=196, a^{2}-2 a b+b^{2}=4, \Rightarrow a+b=14, a-b=2$ and, consequently, $a=8$ (cm), $b=6$ (cm). Since $S=p r$, then $24=12 r, r=2$ (cm).

Answer: $2 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0767.jpg?height=507&width=521&top_left_y=881&top_left_x=97)

Fig. 10.78