SAT Question 603c18ad: Right Triangles And Trigonometry answer and explanation with Desmos solution

Choice D is correct. It's given that in triangle $XYZ$, the measure of angle $X$ is $90^{\circ }$, so triangle $XYZ$ is a right triangle. It’s also given that point $W$ lies on segment $YZ$, and segment $WX$ is perpendicular to segment $YZ$. It follows that triangle $WYX$ is a right triangle. Triangle $WYX$ and triangle $XYZ$ are right triangles with the same interior angle measure at point $Y$. It follows that triangle $WYX$ and triangle $XYZ$ are similar, where angles $W$, $Y$, and $X$ in triangle $WYX$ correspond to angles $X$, $Y$, and $Z$, respectively, in triangle $XYZ$. Since corresponding angles in similar triangles are congruent, their tangents have equal value. Thus, the tangent of angle $Z$ is equal to the tangent of angle $X$ in triangle $WYX$. The opposite side of angle $X$ in triangle $WYX$ is segment $WY$, and the adjacent side is segment $WX$. It’s given that the length of segment $WY$ is $572$ and the length of segment $WX$ is $429$. It follows that the tangent of angle $X$ in triangle $WYX$ is $\frac{572}{429}$, or $\frac{4}{3}$. Therefore, the value of $\tan Z$ is $\frac{4}{3}$.