Question 4
Plot points $A$, $B$, $C$ at coordinates $(0,0)$, $(0,1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $AB$ and $BC$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise $45$ degrees about point $A$. Let $X_{2}$ be the area swept out when Calvin rotates $S$ clockwise $45$ degrees about point $A$. Find $\frac{X_{1}+X_{2}}{2}$.
Supplemental illustration (AI-generated):
Supplemental illustration (AI-generated):