Difference between revisions of "2021 AMC 12B Problems/Problem 5"
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~IceMatrix | ~IceMatrix | ||
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+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/DvpN56Ob6Zw?t=776 | ||
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+ | ~Interstigation | ||
==See Also== | ==See Also== |
Revision as of 01:09, 23 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 Punxsutawney Phil
https://youtube.com/watch?v=qpvS2PVkI8A&t=335s
Video Solution by OmegaLearn (Rotation & Reflection tricks)
~ pi_is_3.14
Video Solution by Hawk Math
https://www.youtube.com/watch?v=VzwxbsuSQ80
Video Solution by TheBeautyofMath
https://youtu.be/GYpAm8v1h-U?t=860 (for AMC 10B)
https://youtu.be/EMzdnr1nZcE?t=814 (for AMC 12B)
~IceMatrix
Video Solution by Interstigation
https://youtu.be/DvpN56Ob6Zw?t=776
~Interstigation
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 Problem 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.