The altitude to the hypotenuse of a $30-60-90^\circ$ right triangle is divided into two segments of lengths $x < y$ by the median to the shortest side of the triangle. What is the ratio $\tfrac{x}{x+y}$?
$$\textbf{(A)}~\frac{4}{9} \qquad \textbf{(B)}~\frac{\sqrt3}{4} \qquad \textbf{(C)}~\frac{4}{9} \qquad \textbf{(D)}~\frac{5}{11} \qquad \textbf{(E)}~\frac{4\sqrt3}{15}$$
Difficulty: 45/100 - A relatively challenging geometry problem that allows many different approaches
Core Concepts: areas, similarity, mass points, coordinates, cevians, ratios
Challenges: Competitors who are not experienced in ratio finding through techniques like mass points or areas may be flustered.
Techniques: Many different approaches are viable. The most straightforward comes from coordinates, though easier solutions are provided by mass points, similar triangles, or areas.
Error-prone Steps: N/A
Ideal Time:
Experienced: ≤ 2 min
Intermediate: ≤ 3-5 min
Beginner: ≤ 8-10 min