Rectangles ABCD and EFGH are similar. The length of each side of EFGH is 6 times

Choice D is correct. The area of a rectangle is given by $bh$, where $b$ is the length of the base of the rectangle and $h$ is its height. Let $x$ represent the length, in units, of the base of rectangle $\mathit{\text{ABCD}}$, and let $y$ represent its height, in units. Substituting $x$ for $b$ and $y$ for $h$ in the formula $bh$ yields $xy$. Therefore, the area, in square units, of $\mathit{\text{ABCD}}$ can be represented by the expression $xy$. It’s given that the length of each side of $\mathit{\text{EFGH}}$ is $6$ times the length of the corresponding side of $\mathit{\text{ABCD}}$. Therefore, the length, in units, of the base of $\mathit{\text{EFGH}}$ can be represented by the expression $6x$, and its height, in units, can be represented by the expression $6y$. Substituting $6x$ for $b$ and $6y$ for $h$ in the formula $bh$ yields $\left(6x\right)\left(6y\right)$, which is equivalent to $36xy$. Therefore, the area, in square units, of $\mathit{\text{EFGH}}$ can be represented by the expression $36xy$. It’s given that the area of $\mathit{\text{ABCD}}$ is $54$ square units. Since $xy$ represents the area, in square units, of $\mathit{\text{ABCD}}$, substituting $54$ for $xy$ in the expression $36xy$ yields $36\left(54\right)$, or $\mathrm{1,944}$. Therefore, the area, in square units, of $\mathit{\text{EFGH}}$ is $\mathrm{1,944}$.

Choice A is incorrect. This is the area of a rectangle where the length of each side of the rectangle is $\sqrt{\frac{1}{6}}$, not $6$, times the length of the corresponding side of $\mathit{\text{ABCD}}$.

Choice B is incorrect. This is the area of a rectangle where the length of each side of the rectangle is $\sqrt{\frac{2}{3}}$, not $6$, times the length of the corresponding side of $\mathit{\text{ABCD}}$.

Choice C is incorrect. This is the area of a rectangle where the length of each side of the rectangle is $\sqrt{6}$, not $6$, times the length of the corresponding side of $\mathit{\text{ABCD}}$.