The angle formed by the lines $d_{1}$ and $d_{2}$ given above measures $30^{\circ}$. First, a perpendicular $A_{1}B_{1}$ is drawn from point $A_{1}$ on line $d_{1}$ to line $d_{2}$. Then, a perpendicular $B_{1}A_{2}$ is drawn from point $B_{1}$ to line $d_{1}$, and a perpendicular $A_{2}B_{2}$ is drawn from the foot of the perpendicular $A_{2}$ to line $d_{2}$, and this process continues. Given that $|A_{1}B_{1}| = 12$ cm, what is the sum of the lengths of all perpendiculars drawn to line $d_{2}$ in this manner, $|A_{1}B_{1}| + |A_{2}B_{2}| + |A_{3}B_{3}| + \cdots$, in cm? A) 32 B) 36 C) 38 D) 40 E) 48
The angle formed by the lines $d_{1}$ and $d_{2}$ given above measures $30^{\circ}$. First, a perpendicular $A_{1}B_{1}$ is drawn from point $A_{1}$ on line $d_{1}$ to line $d_{2}$. Then, a perpendicular $B_{1}A_{2}$ is drawn from point $B_{1}$ to line $d_{1}$, and a perpendicular $A_{2}B_{2}$ is drawn from the foot of the perpendicular $A_{2}$ to line $d_{2}$, and this process continues.
Given that $|A_{1}B_{1}| = 12$ cm, what is the sum of the lengths of all perpendiculars drawn to line $d_{2}$ in this manner, $|A_{1}B_{1}| + |A_{2}B_{2}| + |A_{3}B_{3}| + \cdots$, in cm?
A) 32\\
B) 36\\
C) 38\\
D) 40\\
E) 48