jmc

First notice that this is a right triangle, so two of the altitudes are the legs, whose lengths are $15$ and $20$. The third altitude, whose length is $x$, is the one drawn to the hypotenuse. The area of the triangle is $\frac{1}{2}(15)(20)=150$. Using 25 as the base and $x$ as the altitude, we have $$\frac{1}{2}(25)x=150,\text{so}x=\frac{300}{25}=\boxed{12}.$$



$$\text{OR}$$

Since the three right triangles in the figure are similar, $$\frac{x}{15}=\frac{20}{25},\text{so}x=\frac{300}{25}=\boxed{12}.$$