1991 AMC 12 Problem 10

Find the shortest and longest chord through $P$

Except at the extreme lengths, each attainable chord length occurs in two directions

The longest chord through $P$ is a diameter of length $30.$ The shortest is perpendicular to the radius through $P,$ and has length $$2\sqrt{15^2-9^2}=24.$$ As the chord rotates between these positions, its length varies continuously from $24$ to $30.$ The endpoint lengths each occur once, while each of the five interior integer lengths $25,26,27,28,29$ occurs twice. Thus the count is $1+2(5)+1=12.$

Thus the correct answer is B.