The ratio of the length of line segment X Y to the length of line segment

Choice A is correct. It’s given that the ratio of the length of line segment $XY$ to the length of line segment $ZV$ is $6$ to $1$, which means $\frac{XY}{ZV}=\frac{6}{1}$. It’s given that the length of line segment $XY$ is $102$ inches. If the length, in inches, of line segment $ZV$ is represented by $\mathcal{l}$, the value of $\mathcal{l}$ can be calculated by solving the equation $\frac{102}{\mathcal{l}}=\frac{6}{1}$, or $\frac{102}{\mathcal{l}}=6$. Multiplying each side of this equation by $\mathcal{l}$ yields $102=6\mathcal{l}$. Dividing each side of this equation by $6$ yields $17=\mathcal{l}$. Therefore, the length of line segment $ZV$ is $17$ inches.

Choice B is incorrect. This is the length, in inches, of line segment $ZV$ if the length of line segment $XY$ is $576$, not $102$, inches.

Choice C is incorrect. This is the length, in inches, of line segment $XY$, not line segment $ZV$.

Choice D is incorrect. This is the length, in inches, of line segment $ZV$ if the ratio of the length of line segment $XY$ to the length of line segment $ZV$ is $1$ to $6$, not $6$ to $1$.