imc

As the problem has no diagram, we draw a diagram. The hypotenuse has length $29$. Let $P$ be the foot of the altitude from $B$ to $AC$. Note that $BP$ is the shortest possible length of any segment. Writing the area of the triangle in two ways, we can solve for $BP=\frac{20\cdot 21}{29}$, which is between $14$ and $15$. Let the line segment be $BX$, with $X$ on $AC$. As you move $X$ along the hypotenuse from $A$ to $P$, the length of $BX$ strictly decreases, hitting all the integer values from $20,19,\dots 15$ (IVT). Similarly, moving $X$ from $P$ to $C$ hits all the integer values from $15,16,\dots ,21$. This is a total of $\boxed{13}$ distinct line segments. (asymptote diagram added by elements2015)