Solve the triangle whose area $t=357.18 \mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$.


If $2 s=a+b+c$, then the area of the triangle is:

$$
t=\sqrt{s \cdot(s-a) \cdot(s-b) \cdot(s-c)}
$$

Since

$$
a: b: c=4: 5: 6
$$

therefore

$$
b=\frac{5 a}{4}, c=\frac{6 a}{4} \text { and } s=\frac{15 a}{8}
$$

Substituting these values into the area formula, we get:

$$
357.18=\frac{15 a^{2}}{64} \sqrt{7}
$$

from which

$$
a=\sqrt{\frac{357.18 \times 64}{15 \sqrt{7}}}=24 \text{ cm}
$$

furthermore

$$
b=30 \text{ cm}, \quad c=36 \text{ cm}
$$

The angles can be most conveniently obtained from the formulas

$$
\operatorname{tg}=\frac{\alpha}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}
$$

etc. Performing the calculations, we get:

$$
\alpha=41^{\circ} 24^{\prime} 34^{\prime \prime}, \beta=55^{\circ} 46^{\prime} 16^{\prime \prime}, \gamma=82^{\circ} 49^{\prime} 10^{\prime \prime}
$$

(Andor Szenes, Kaposvár.)

Number of solutions: 53.