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Geometry and Trigonometry Difficulty: Easy

	Perimeter
Triangle $T U V$	$50$
Triangle $X Y Z$	$150$

The table gives the perimeters of similar triangles $T U V$ $T U V$ and $X Y Z$ $X Y Z$ , where $T U$ corresponds to $X Y$ . The length of $T U$ is $6$ .

What is the length of $X Y$ ?

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Explanation

Choice C is correct. It’s given that triangle $T U V$ is similar to triangle $X Y Z$ , and $\bar{T U}$ corresponds to $\bar{X Y}$ . If two triangles are similar, then the ratio of their perimeters is equal to the ratio of their corresponding sides. It’s given that the perimeter of triangle $T U V$ is $50$ , the perimeter of triangle $X Y Z$ is $150$ , and the length of $\bar{T U}$ is $6$ . Let $n$ represent the length of $\bar{X Y}$ . It follows that $\frac{50}{150} = \frac{6}{n}$ , or $\frac{1}{3} = \frac{6}{n}$ . Multiplying each side of this equation by $n$ yields $\frac{n}{3} = 6$ . Multiplying each side of this equation by $3$ yields $n equals 18$ . Therefore, the length of $\bar{X Y}$ is $18$ .

Choice A is incorrect. This is the solution to $\frac{3}{1} = \frac{6}{n}$ , not $\frac{1}{3} = \frac{6}{n}$ .

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

Choice D is incorrect. This is the sum of the length of $\bar{T U}$ and the perimeter of triangle $T U V$ , not the length of $\bar{X Y}$ .