In triangle $ABC$ if the median to side $BC$ has length $\left[11 - 6\sqrt{3}\right]^{\frac{1}{2}}$ and it divides angle $\angle A$ into angles $30°$ and $45°$. Then length of side $BC$ is __________.
Step-by-Step Solution
Key Concept: The median configuration with specified angle inclinations and centroid properties uniquely determines vertex positions through trigonometric constraints.
With $O$ as origin and $x$-axis as median, point $A$ is at $(x_1, y_1)$ where $\frac{y_1}{x_1} = \tan 30° = \frac{1}{\sqrt{3}}$. Point $B$ satisfies $\frac{y_2}{x_2} = \tan 135° = -1$, giving $y_1 = \frac{x_1}{\sqrt{3}}$ and $y_2 = -x_2$. With centroid condition $\frac{y_1+y_2}{2} = 0$ and median properties, we get $x_1 = \frac{2\sqrt{3}p}{\sqrt{3}+1}$, $x_2 = \frac{2p}{\sqrt{3}+1}$, $y_1 = \frac{2p}{\sqrt{3}+1}$, $y_2 = -\frac{2p}{\sqrt{3}+1}$, and $BC = 2$.
Correct Answer: 2