Difference between revisions of "2021 AMC 12B Problems/Problem 5"
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| − | {{AMC10 box|year=2021|ab=B|before= | + | {{AMC10 box|year=2021|ab=B|before=num-b=8|num-a=10}} |
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Revision as of 23:40, 11 February 2021
Contents
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 
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 
.
--abhinavg0627
Video Solution by OmegaLearn (Rotation & Reflection tricks)
~ pi_is_3.14
See Also
| 2021 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 4 |
Followed by Problem 6 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
| 2021 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by num-b=8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
