29. In a scalene triangle $ABC$, the lengths of two altitudes are 4 and 12. If the length of the third altitude is also an integer, what is the maximum it can be?
(A) 4 ;
(B) 5 ;
(C) 6 ;
(D) 7 ;
(E) None of the above.
$\left\lvert\, \frac{29}{\mathrm{~B}}\right.$
29. Suppose the length of the base on $a$ is $4$, the height on $b$ is $12$, and the height on $c$ is the unknown $h$. Let $K$ be the area of $\triangle A B C$, then $4 a=12 b=h c=2 K$. By the triangle inequality $c<a+b$, in other words,
$$
\frac{2 K}{h}\frac{2 K}{4}-\frac{2 K}{12},
$$

they are equivalent to
$$
\frac{1}{h}\frac{1}{4}-\frac{1}{12}=\frac{1}{6} .
$$

So $3<h<6$, if $h$ is an integer, it cannot exceed 5.