Question

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$ -axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself?

$359$

$360$

$719$

$720$

$721$

Solution:

Since we are working with angles and reflections, working with polar coordinates would make this problem easier to deal with.

Let $(r, \theta)$ be a polar coordinate. Rotating this by $k$ degrees counterclockwise maps the point to $(r, \theta + k^{\circ})$ and then reflecting it maps it to $(r, 180 - \theta - k^{\circ}.)$

Therefore, we have that $T_k(r, \theta) = (r, 180 - \theta - k^{\circ}).$

From this, we can see that $T_{k + 1}(T_k(r, \theta)) =$ $T_{k + 1}(r, 180^{\circ} - \theta - k^{\circ}) =$ $(r, \theta - 1^{\circ}).$

Now, let's analyze what happens to the point $(1, 0^{\circ}).$

After $T_1,$ we get $(1, 179^{\circ}).$

After $T_2,$ we get $(1, -1^{\circ}).$

After $T_3,$ we get $(1, 178^{\circ}).$

After $T_4,$ we get $(1, -2^{\circ}).$

$\vdots$

After $T_{2n - 1},$ we get $(1, 180^{\circ} - k^{\circ}).$

After $T_{2n},$ we get $(1, -k^{\circ}).$

From this, we can see that the first time the angle is back to $0^{\circ}$ is when $n = 180$ and $k = 359.$

Thus, A is the correct answer.

2022 AMC 10A Problem 18

Solution:

Problem 18 in Other Years