Question

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 $90^\circ$ 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?

$1$

$2$

$3$

$4$

$5$

Solution:

The dilation just scales about the origin, so it commutes with both the rotation and the reflection. That's $2$ pairs. The translation commutes with the reflection across the $x$ -axis too, since either order sends $(x, y) \to (x + 2, -y).$ The other three pairs fail: the translation clashes with the rotation and with the dilation, and the rotation clashes with the reflection. So $3$ pairs commute. Thus, C is the correct answer.

2024 AMC 10A Problem 13

Solution:

Problem 13 in Other Years