2020 AMC 12B Problems/Problem 19
Square 
in the coordinate plane has vertices at the points 
and 
Consider the following four transformations: 
a rotation of 
counterclockwise around the origin; 
a rotation of clockwise around the origin; 
a reflection across the 
-axis; and 
a reflection across the 
-axis.
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying 
and then 
would send the vertex 
at 
to 
and would send the vertex 
at 
to itself. How many sequences of 
transformations chosen from 
will send all of the labeled vertices back to their original positions? (For example, 
is one sequence of 
transformations that will send the vertices back to their original positions.)
Solution
Hopefully, someone will think of a better one, but here is an indirect answer, use only if you are really desperate.
moves can be made, and each move have choices, so a total of 
moves. First, after the moves, Point A can only be in first quadrant or third quadrant . Only the one in the first quadrant works, so divide by 
. Now, C must be in the opposite quadrant as A. B can be either in the second (
) or fourth quadrant (
) , but we want it to be in the second quadrant, so divide by again. Now as A and B satisfy the conditions, C and D will also be at their original spot. 
. The answer is 
~Kinglogic
| 2020 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 18 |
Followed by Problem 20 |
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