Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2024 AMC 10A Problems/Problem 13"

m (Protected "2024 AMC 10A Problems/Problem 13" ([Edit=Allow only administrators] (expires 04:59, 8 November 2024 (UTC)) [Move=Allow only administrators] (expires 04:59, 8 November 2024 (UTC))) [cascading])
Line 1: Line 1:
 +
== 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?
 +
 +
 +
<math>\textbf{(A)1}\qquad\textbf{(B)2}\qquad\textbf{(C)3}\qquad\textbf{(D)4}\qquad\textbf{(E)5}</math>
 +
 +
== Solution 1 ==
 +
Note that the dilation with any of the other three transformations are commute except with the <math>2</math> right translation.
 +
 +
The reflection across the <math>x-</math>axis and <math>90^{\circ}</math> rotation both are not commute when paired with the translation <math>2</math> right (Consider the point <math>(1,0)</math> and visualize), and the reflection across the <math>x-</math>axis and the <math>90^{\circ}</math> rotation are not commute together (Again, visualize the point <math>(1,0)</math> Therefore, the answer is <math>\boxed{\text{(C) }2}</math>

Revision as of 16:58, 8 November 2024

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 $\boxed{\text{(C) }2}$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2024_AMC_10A_Problems/Problem_13&oldid=231939"