SAT Question 55bb437a: Right Triangles And Trigonometry answer and explanation with Desmos solution

Since $\tan B=\frac{3}{4}$, triangle $ABC$ and triangle $DBE$ are both similar to $3$-$4$-$5$ triangles. This means that they are both similar to the right triangle with sides of lengths $3$, $4$, and $5$. Since $BC=15$, which is $3$ times as long as the hypotenuse of the $3$-$4$-$5$ triangle, the similarity ratio of triangle $ABC$ to the $3$-$4$-$5$ triangle is $3:1$. Therefore, the length of $\overline{AC}$ (the side opposite to $\angle B$) is $3\times 3=9$, and the length of $\overline{AB}$ (the side adjacent to $\angle B$) is $4\times 3=12$. It is also given that $DA=4$. Since $AB=DA+DB$ and $AB=12$, it follows that $DB=8$, which means that the similarity ratio of triangle $DBE$ to the $3$-$4$-$5$ triangle is $2:1$ ($\overline{DB}$ is the side adjacent to $\angle B$). Therefore, the length of $\overline{DE}$, which is the side opposite to $\angle B$, is $3\times 2=6$.