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

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 $(1,0)$

$1. W$ and $X$ . Going $W\rightarrow X$ ends with the point $(0,3)$ . Going $X\rightarrow W$ ends in the point $(1,2)$ , so this pair does not work

$2.

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
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2024 AMC 10A Problems/Problem 13

Contents

Problem

Solution 1

Solution 1

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