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Geometry and Trigonometry / Area and volume Difficulty: Medium

	Perimeter
Triangle $T U V$	$37$
Triangle $X Y Z$	$333$

The table gives the perimeters of similar triangles $T U V$ and $X Y Z$ , where $T U$ corresponds to $X Y$ . The length of $T U$ is $18$ .

What is the length of $X Y$ ?

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Explanation

Choice D is correct. It's given that triangle $X Y Z$ is similar to triangle $T U V$ . Therefore, each side of triangle $X Y Z$ is $k$ times its corresponding side of triangle $T U V$ , where $k$ is a constant. It follows that the perimeter of triangle $X Y Z$ is $k$ times the perimeter of triangle $T U V$ . It's also given that $\bar{T U}$ corresponds to $\bar{X Y}$ and the length of $\bar{T U}$ is $18$ . Let $x$ represent the length of $\bar{X Y}$ . It follows that $x equals 18 k$ . The table shows that the perimeters of triangles $T U V$ and $X Y Z$ are $37$ and $333$ , respectively. It follows that $333 equals 37 k$ , or $9 equals k$ . Substituting $9$ for $k$ in the equation $x equals 18 k$ yields $x equals left parenthesis 18 right parenthesis left parenthesis 9 right parenthesis$ , or $x equals 162$ . Therefore, the length of $\bar{X Y}$ is $162$ .

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

Choice B is incorrect. This is the length of $\bar{T U}$ , not the length of $\bar{X Y}$ .

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