If the base of the triangle is $2 x$, then its height is

$$m=\sqrt{625-x^{2}}$$

and its area is

$$t=m x=x \sqrt{625-x^{2}}=168$$

or

$$x^{2}\left(626-x^{2}\right)=168^{2}$$

Rearranging,

$$x^{4}-625 x^{2}+168^{2}=0$$

from which

$$x_{1}^{2}=576 ; x_{2}^{2}=49$$

thus the positive roots are:

$$x_{1}=24 ; x_{2}=7$$

Therefore, the base of the triangle is either $48 \mathrm{~cm}$ or $14 \mathrm{~cm}$.