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Geometry and Trigonometry Difficulty: Hard

In the figure shown, $\bar{W Z}$ and $\bar{X Y}$ intersect at point $Q$ . $upper Y upper Q equals 63$ , $upper W upper Q equals 70$ , $upper W upper X equals 60$ , and $upper X upper Q equals 120$ . What is the length of $\bar{Y Z}$ ?

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Explanation

The correct answer is $54$ . The figure shown includes two triangles, triangle $W Q X$ and triangle $Y Q Z$ , such that angle $W Q X$ and angle $Y Q Z$ are vertical angles. It follows that angle $W Q X$ is congruent to angle $Y Q Z$ . It’s also given in the figure that the measures of angle $W$ and angle $Y$ are $a °$ . Therefore angle $W$ is congruent to angle $Y$ . Since triangle $W Q X$ and triangle $Y Q Z$ have two pairs of congruent angles, triangle $W Q X$ is similar to triangle $Y Q Z$ by the angle-angle similarity postulate, where $\bar{Y Z}$ corresponds to $\bar{W X}$ , and $\bar{Y Q}$ corresponds to $\bar{W Q}$ . Since the lengths of corresponding sides in similar triangles are proportional, it follows that $\frac{Y Z}{W X} = \frac{Y Q}{W Q}$ . It’s given that $upper Y upper Q equals 63$ , $upper W upper Q equals 70$ , and $upper W upper X equals 60$ . Substituting $63$ for $Y Q$ , $70$ for $W Q$ , and $60$ for $W X$ in the equation $\frac{Y Z}{W X} = \frac{Y Q}{W Q}$ yields $\frac{Y Z}{60} = \frac{63}{70}$ . Multiplying each side of this equation by $60$ yields $Y Z = (\frac{63}{70}) (60)$ , or $upper Y upper Z equals 54$ . Therefore, the length of $\bar{Y Z}$ is $54$ .