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Geometry and Trigonometry Difficulty: Medium

Triangle $A B C$ is similar to triangle $X Y Z$ , where $A$ , $B$ , and $C$ correspond to $X$ , $Y$ , and $Z$ , respectively. In triangle $A B C$ , the length of $\bar{A B}$ is $170$ and the length of $\bar{B C}$ is $850$ . In triangle $X Y Z$ , the length of $\bar{Y Z}$ is $60$ . What is the length of $\bar{X Y}$ ?

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Explanation

Choice D is correct. It's given that triangle $A B C$ is similar to triangle $X Y Z$ , where $A$ , $B$ , and $C$ correspond to $X$ , $Y$ , and $Z$ , respectively. It follows that side $A B$ corresponds to side $X Y$ and side $B C$ corresponds to side $Y Z$ . Since the lengths of corresponding sides in similar triangles are proportional, it follows that $\frac{X Y}{A B} = \frac{Y Z}{B C}$ . Substituting $170$ for $A B$ , $60$ for $Y Z$ , and $850$ for $B C$ in this equation yields $\frac{X Y}{170} = \frac{60}{850}$ . Multiplying each side of this equation by $170$ yields $upper X upper Y equals 12$ . Therefore, the length of $\bar{X Y}$ is $12$ .

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

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

Choice C is incorrect. This is the length of $\bar{Y Z}$ , not $\bar{X Y}$ .