In a right-angled triangle, the height drawn to the hypotenuse is $12 \mathrm{~cm}$; the difference between the two segments of the hypotenuse is $7 \mathrm{~cm}$. What are the lengths of the sides of the triangle?
The two segments of the hypotenuse are $x$ and $x-7$, so

$$
x(x-7)=12^{2}
$$

or

$$
x^{2}-7 x-144=0
$$

from which the positive root is

$$
x=16
$$

One segment of the hypotenuse is $16 \mathrm{~cm}$, the other is $9 \mathrm{~cm}$, so one leg is

$$
\sqrt{9(9+16)=15} \mathrm{~cm}
$$

and the other is

$$
\sqrt{16(9+16)=20} \mathrm{~cm}
$$

(László Szántó, Pécs.)