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Geometry and Trigonometry Difficulty: Hard

Rectangles $ABCD$ and $EFGH$ are similar. The length of each side of $EFGH$ is $6$ times the length of the corresponding side of $ABCD$ . The area of $ABCD$ is $54$ square units. What is the area, in square units, of $EFGH$ ?

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Explanation

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

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 $ABCD$ .

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 $ABCD$ .

Choice C is incorrect. This is the area of a rectangle where the length of each side of the rectangle is $StartRoot 6 EndRoot$ , not $6$ , times the length of the corresponding side of $ABCD$ .