2024 AMC 10A Problems/Problem 13
Contents
Problem
Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:
• a translation 2 units to the right,
• a 90°- rotation counterclockwise about the origin,
• a reflection across the 𝑥-axis, and
• a dilation centered at the origin with scale factor 2 .
Of the 6 pairs of distinct transformations from this list, how many commute?
Solution 1
Examine
Solution 1
Label the transformations as follows:
• a translation 2 units to the right (W)
• a 90°- rotation counterclockwise about the origin (X)
• a reflection across the 𝑥-axis (Y)
• a dilation centered at the origin with scale factor 2 (Z)
Now, examine each possible pair of transformations with the point 
and 
. Going 
ends with the point 
. Going 
ends in the point 
, so this pair does not work
$2.
See also
| 2024 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
