Choice B is correct. The sine of an acute angle in a right triangle is the ratio of the length of the side opposite that angle to the length of the hypotenuse. The hypotenuse of a right triangle is the side opposite the right angle. In right triangle $A B C$ , side $B C$ is the side opposite angle $A$ and side $A B$ is the hypotenuse. It's given that the length of side $B C$ is $35$ units and the length of side $A B$ is $171$ units. Therefore, the value of $\sin A$ is $StartFraction 35 Over 171 EndFraction$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the ratio of the length of the hypotenuse to the length of the side opposite angle $A$ rather than the ratio of the length of the side opposite angle $A$ to the length of the hypotenuse.

Choice D is incorrect. This is the length of the hypotenuse rather than $\sin A$ .