2024 AMC 10A Problems/Problem 13

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?

$\textbf{(A)1}\qquad\textbf{(B)2}\qquad\textbf{(C)3}\qquad\textbf{(D)4}\qquad\textbf{(E)5}$

Solution 1

Note that the dilation with any of the other three transformations are commute except with the $2$ right translation.

The reflection across the $x-$ axis and $90^{\circ}$ rotation both are not commute when paired with the translation $2$ right (Consider the point $(1,0)$ and visualize), and the reflection across the $x-$ axis and the $90^{\circ}$ rotation are not commute together (Again, visualize the point $(1,0)$ Therefore, the answer is $\boxed{\text{(C) }2}$

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2024 AMC 10A Problems/Problem 13

Problem

Solution 1