Problem:
The point in the -plane is first rotated counterclockwise by around the point and then reflected about the line . The image of after these two transformations is at . What is ?
Answer Choices:
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Solution:
The final image of is . We know the reflection rule for reflecting over is . So before the reflection and after rotation the point is .
By definition of rotation, the slope between and must be perpendicular to the slope between and . The first slope is . This means the slope of and is .
Rotations also preserve distance to the center of rotation, and since we only "travelled" up and down by the slope once to get from to it follows we shall only use the slope once to travel from to .
Therefore point is located at . The answer is .
Let us reconstruct that coordinate plane as the complex plane. Then, the point becomes . A rotation around the point can be done by translating the point to the origin, rotating around the origin by , and then translating the origin back to the point .
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By basis reflection rules, the reflection of about the line is . Hence, we have
from which .
The problems on this page are the property of the MAA's American Mathematics Competitions