10.223. The perimeter of a right-angled triangle is 60 cm. Find its sides if the height drawn to the hypotenuse is 12 cm.
## Solution.

Let $a$ and $b$ be the legs, and $c$ be the hypotenuse of the triangle. Since $a+b+c=60$, then $a+b=60-c$, from which $(a+b)^{2}=(60-c)^{2}$ or $a^{2}+2 a b+b^{2}=3600-120 c+c^{2} \quad\left(^{*}\right.$ ). But $a^{2}+b^{2}=c^{2}$, and $6 c=0.5 a b$ (the area of the triangle). Substituting these expressions into the equation ( ${ }^{*}$ ), we get $c^{2}+24 c=3600-120 c+c^{2}, 144 c=3600$, i.e., $c=25$ cm. We have the system of equations $\left\{\begin{array}{l}a+b=35, \\ a b=300,\end{array}\right.$ i.e., $a$ and $b$ are the roots of the equation $x^{2}-35 x+300=0$. Therefore, $a_{1}=20, a_{2}=15 ; b_{1}=15, b_{2}=20$.

Answer: 15, 20, and 25 cm.