2018 AMC 10A Problem 16

Find the altitude from $B$ to the hypotenuse

Integer-length segments occur in symmetric pairs around the altitude

Let $P$ be the foot of the altitude from $B$ to $\overline{AC}.$ The Pythagorean Theorem gives $AC=29.$ Computing the area in two ways gives $$\dfrac{29\cdot PB}{2}=\dfrac{20\cdot21}{2},$$ so $PB=\dfrac{420}{29},$ which lies between $14$ and $15.$

As the endpoint moves from $A$ to $P,$ its distance from $B$ decreases continuously from $20$ to $PB.$ Thus there is one segment of each integer length $15,16,17,18,19,20.$ As the endpoint moves from $P$ to $C,$ the distance increases continuously from $PB$ to $21,$ giving one segment of each integer length $15,16,17,18,19,20,21.$ These are $6+7=13$ distinct segments.

Thus, D is the correct answer.