SAT Question eeb4143c: Lines, Angles, and Triangles answer and explanation with Desmos solution

The correct answer is either $\frac{10}{3}$, $\frac{15}{4}$, or $\frac{25}{6}$. The area of triangle $ABC$ can be expressed as $\frac{1}{2}\times (5+7)\times y$ or $6y$. It's given that the area of triangle $ABC$ is at least $48$ but no more than $60$. It follows that $48\le 6y\le 60$. Dividing by $6$ to isolate $y$ in this compound inequality yields $8\le y\le 10$. Since $y$ is an integer, $y=8$, $9$, or $10$. In the given figure, the two right triangles shown are similar because they have two pairs of congruent angles: their respective right angles and angle $A$. Therefore, the following proportion is true: $\frac{x}{y}=\frac{5}{12}$. Substituting $8$ for $y$ in the proportion results in $\frac{x}{8}=\frac{5}{12}$. Cross multiplying and solving for $x$ yields $\frac{10}{3}$. Substituting $9$ for $y$ in the proportion results in $\frac{x}{9}=\frac{5}{12}$. Cross multiplying and solving for $x$ yields $\frac{15}{4}$. Substituting $10$ for $y$ in the proportion results in $\frac{x}{10}=\frac{5}{12}$. Cross multiplying and solving for $x$ yields $\frac{25}{6}$. Note that 10/3, 3.333, 15/4, 3.75, 3.750, 25/6, 4.166, and 4.167 are examples of ways to enter a correct answer.