The sides of a right-angled triangle form an arithmetic progression. The area of the triangle is $216 \mathrm{~cm}^{2}$. Calculate the sides and angles of the triangle.
Let $a-d, a, a+d$ be the three sides of the triangle. Then

$$
\begin{gathered}
a^{2}+(a-d)^{2}=(a+d)^{2} \\
a^{2}=4 a d, \quad \text { or } a=4 d
\end{gathered}
$$

thus

$$
a(a-d)=12 d^{2}=432
$$

from which

$$
d=6
$$

Therefore, the three sides of the triangle are:

$$
18 \mathrm{~cm}, 24 \mathrm{~cm} \text {, and } 30 \mathrm{~cm} \text {. }
$$

If $\alpha$ is the angle opposite to $a$, then

$$
\sin \alpha=\frac{a}{a+d}=\frac{4}{5}, \alpha=53^{\circ} 7^{\prime} 48^{\prime \prime} \text { and } \beta=90^{\circ}-\alpha=36^{\circ} 52^{\prime} 12^{\prime \prime}
$$

(Jakab Bauer, Zenta.)