Question 7
A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.
Semicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, which is inscribed in another semicircle $S_{3}$. The radii of $S_{1}$ and $S_{3}$ are $1$ and $10$, respectively, and the diameters of $S_{1}$ and $S_{3}$ are parallel. The endpoints of the diameter of $S_{3}$ are $A$ and $B$, and $S_{2}$'s arc is tangent to $AB$ at $C$. Compute $AC \cdot CB$.
Semicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, which is inscribed in another semicircle $S_{3}$. The radii of $S_{1}$ and $S_{3}$ are $1$ and $10$, respectively, and the diameters of $S_{1}$ and $S_{3}$ are parallel. The endpoints of the diameter of $S_{3}$ are $A$ and $B$, and $S_{2}$'s arc is tangent to $AB$ at $C$. Compute $AC \cdot CB$.