A perpendicular bisector plane intersects the faces of a corner at areas that are in the ratio $16: 21: 28$. What are the lengths of the edges if the diagonal of the bisector plane is $29 \mathrm{~cm}$?
If we denote the edges meeting at a corner by $a$, $b$, and $c$, then according to the problem,

$$
a b: b c=16: 21, \quad \text { or } \quad a=\frac{16}{21} c
$$

and

$$
a b: a c=16: 28, \quad \text { or } \quad b=\frac{4}{7} c
$$

furthermore

$$
a^{2}+b^{2}+c^{2}=29^{2}
$$

or

$$
\left[\left(\frac{16}{21}\right)^{2}+\left(\frac{4}{7}\right)^{2}+1\right] c^{2}=29^{2}
$$

from which the positive value of $c$ is $21 \text{ cm}$, thus

$$
a=16 \text{ cm} \quad \text { and } \quad b=12 \text{ cm} .
$$