Example 12 (2003 National College Entrance Examination Question) Given the four vertices of a rectangle $A(0,0), B(2,0), C(2,1), D(0,1)$. A particle starts from the midpoint $P_{0}$ of $A B$ and moves in a direction making an angle $\theta$ with $A B$ to reach point $P_{1}$ on $B C$, then reflects successively to points $P_{2}, P_{3}$, and $P_{4}$ on $C D, D A$, and $A B$ (the angle of incidence equals the angle of reflection). If $P_{4}$ coincides with $P_{0}$, then $\tan \theta=(\quad)$.
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{1}{2}$
D. 1
Solution: Choose C. Reason: According to the problem, it can be solved step by step, but there are too many steps and the calculations are too complex. In fact, by applying an axis reflection transformation as shown in Figure 3-5 (folding four rectangles, each with a length of 2 and a width of 1), then \( C P^{\prime}{ }_{2}=C P_{2}, D_{1} P^{\prime}{ }_{3}=D P_{3} \), \( A_{1} P^{\prime}{ }_{4}=A P_{1}\left(A P_{0}\right) \), it is easy to see that \( P_{0}, P_{1}, P_{2}, P_{3}, P_{4} \) lie on a straight line. According to the problem, the coordinates of point \( P_{0} \) are \( (1,0) \), and the coordinates of \( P_{4} \) are \( (5,2) \). Thus, we can find that \( \tan \theta = \frac{2-0}{5-1} = \frac{1}{2} \).