Difference between revisions of "2020 AMC 10A Problems/Problem 23"
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== Solution == | == Solution == | ||
| − | First, any combination of motions we can make must | + | First, any combination of motions we can make must reflect <math>T</math> an even number of times. This is because every time we reflect <math>T</math>, it changes orientation. Once <math>T</math> has been flipped once, no combination of rotations will put it back in place because it is the mirror image; however, flipping it again changes it back to the original orientation. Since we are only allowed <math>3</math> transformations and an even number of them must be reflections, we either reflect <math>T</math> <math>0</math> times or <math>2</math> times. |
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| + | [b]Case 1: 0 reflections on T[/b] | ||
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| + | In this case, we must use <math>3</math> rotations to return <math>T</math> to its original position. Notice that our set of rotations, <math>\{90^\circ,180^\circ,270^\circ\}</math>, contains every multiple of <math>90^\circ</math> except for <math>0^\circ</math>. We can start with any two rotations <math>a,b</math> in <math>\{90^\circ,180^\circ,270^\circ\}</math> and there must be exactly one <math>c \congruent -a - b \pmod{360^\circ}</math> such that we can use the three rotations <math>(a,b,c)</math> which | ||
==See Also== | ==See Also== | ||
Revision as of 22:02, 31 January 2020
Problem
Let 
be the triangle in the coordinate plane with vertices 
and 
Consider the following five isometries (rigid transformations) of the plane: rotations of 
and 
counterclockwise around the origin, reflection across the 
-axis, and reflection across the 
-axis. How many of the 
sequences of three of these transformations (not necessarily distinct) will return to its original position? (For example, a 
rotation, followed by a reflection across the -axis, followed by a reflection across the -axis will return to its original position, but a 
rotation, followed by a reflection across the -axis, followed by another reflection across the -axis will not return to its original position.)
Solution
First, any combination of motions we can make must reflect an even number of times. This is because every time we reflect , it changes orientation. Once has been flipped once, no combination of rotations will put it back in place because it is the mirror image; however, flipping it again changes it back to the original orientation. Since we are only allowed 
transformations and an even number of them must be reflections, we either reflect 
times or 
times.
[b]Case 1: 0 reflections on T[/b]
In this case, we must use rotations to return to its original position. Notice that our set of rotations, 
, contains every multiple of 
except for 
. We can start with any two rotations 
in and there must be exactly one $c \congruent -a - b \pmod{360^\circ}$ (Error compiling LaTeX. Unknown error_msg) such that we can use the three rotations 
which
See Also
| 2020 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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 | ||
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