Difference between revisions of "2020 AMC 10A Problems/Problem 23"
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| − | Let <math>T</math> be the triangle in the coordinate plane with vertices <math>(0,0), (4,0),</math> and <math>(0,3).</math> Consider the following five isometries (rigid transformations) of the plane: rotations of <math>90^{\circ}</math> | + | Let <math>T</math> be the triangle in the coordinate plane with vertices <math>(0,0), (4,0),</math> and <math>(0,3).</math> Consider the following five isometries (rigid transformations) of the plane: rotations of <math>90^{\circ}, 180^{\circ},</math> and <math>270^{\circ}</math> counterclockwise around the origin, reflection across the <math>x</math>-axis, and reflection across the <math>y</math>-axis. How many of the <math>125</math> sequences of three of these transformations (not necessarily distinct) will return <math>T</math> to its original position? (For example, a <math>180^{\circ}</math> rotation, followed by a reflection across the <math>x</math>-axis, followed by a reflection across the <math>y</math>-axis will return <math>T</math> to its original position, but a <math>90^{\circ}</math> rotation, followed by a reflection across the <math>x</math>-axis, followed by another reflection across the <math>x</math>-axis will not return <math>T</math> to its original position.) |
<math>\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 25</math> | <math>\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 25</math> | ||
Revision as of 22:44, 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
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
