Difference between revisions of "2024 AMC 10A Problems/Problem 13"
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<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5</math> | <math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5</math> | ||
− | == Solution 1 ( | + | == Solution 1 (Generalized) == |
Label the given transformations <math>T_1, T_2, T_3,</math> and <math>T_4,</math> respectively. The rules of transformations are: | Label the given transformations <math>T_1, T_2, T_3,</math> and <math>T_4,</math> respectively. The rules of transformations are: | ||
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Applying <math>T_4</math> and then <math>T_2</math> gives <math>(x,y)\to(2x,2y)\to(-2y,2x).</math> <p> | Applying <math>T_4</math> and then <math>T_2</math> gives <math>(x,y)\to(2x,2y)\to(-2y,2x).</math> <p> | ||
Therefore, <math>T_2</math> and <math>T_4</math> commute. | Therefore, <math>T_2</math> and <math>T_4</math> commute. | ||
+ | </li><p> | ||
+ | <li>Applying <math>T_3</math> and then <math>T_4</math> gives <math>(x,y)\to(x,-y)\to(2x,-2y).</math> <p> | ||
+ | Applying <math>T_4</math> and then <math>T_3</math> gives <math>(x,y)\to(2x,2y)\to(2x,-2y).</math> <p> | ||
+ | Therefore, <math>T_3</math> and <math>T_4</math> commute. | ||
</li><p> | </li><p> | ||
</ul> | </ul> | ||
+ | Together, <math>\boxed{\textbf{(C)}~3}</math> pairs of transformations commute: <math>T_1</math> and <math>T_3, T_2</math> and <math>T_4,</math> and <math>T_3</math> and <math>T_4.</math> | ||
+ | |||
+ | ~MRENTHUSIASM | ||
== Solution 2 (Specific) == | == Solution 2 (Specific) == | ||
Label the transformations as follows: | Label the transformations as follows: | ||
− | • a translation 2 units to the right <math>(W)</math> | + | • a translation <math>2</math> units to the right <math>(W)</math> |
− | • a | + | • a <math>90^{\circ}</math>-rotation counterclockwise about the origin <math>(X)</math> |
− | • a reflection across the | + | • a reflection across the <math>x</math>-axis <math>(Y)</math> |
− | • a dilation centered at the origin with scale factor 2 <math>(Z)</math> | + | • a dilation centered at the origin with scale factor <math>2</math> <math>(Z)</math> |
Now, examine each possible pair of transformations with the point <math>(1,0)</math>: | Now, examine each possible pair of transformations with the point <math>(1,0)</math>: | ||
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Label the transformations as follows: | Label the transformations as follows: | ||
− | • a translation 2 units to the right <math>(A)</math> | + | • a translation <math>2</math> units to the right <math>(A)</math> |
− | • a | + | • a <math>90^{\circ}</math>-rotation counterclockwise about the origin <math>(B)</math> |
− | • a reflection across the | + | • a reflection across the <math>x</math>-axis <math>(C)</math> |
− | • a dilation centered at the origin with scale factor 2 <math>(D)</math> | + | • a dilation centered at the origin with scale factor <math>2</math> <math>(D)</math> |
Now, we count each transformation individually. It is not hard to see that <math>AC, BD,</math> and <math>CD</math> are commutative (an easy way to test commutativity for some cases would be to have the original point on the <math>x</math>-axis). | Now, we count each transformation individually. It is not hard to see that <math>AC, BD,</math> and <math>CD</math> are commutative (an easy way to test commutativity for some cases would be to have the original point on the <math>x</math>-axis). |
Revision as of 03:02, 9 November 2024
Contents
[hide]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 units to the right,
- a -rotation counterclockwise about the origin,
- a reflection across the -axis, and
- a dilation centered at the origin with scale factor
Of the pairs of distinct transformations from this list, how many commute?
Solution 1 (Generalized)
Label the given transformations and respectively. The rules of transformations are:
Note that:
- Applying and then gives
Applying and then gives
Therefore, and do not commute.
- Applying and then gives
Applying and then gives
Therefore, and commute. They form a glide reflection.
- Applying and then gives
Applying and then gives
Therefore, and do not commute.
- Applying and then gives
Applying and then gives
Therefore, and do not commute.
- Applying and then gives
Applying and then gives
Therefore, and commute.
- Applying and then gives
Applying and then gives
Therefore, and commute.
Together, pairs of transformations commute: and and and and
~MRENTHUSIASM
Solution 2 (Specific)
Label the transformations as follows:
• a translation units to the right
• a -rotation counterclockwise about the origin
• a reflection across the -axis
• a dilation centered at the origin with scale factor
Now, examine each possible pair of transformations with the point :
and . ends with the point . Going ends in the point , so this pair does not work
and . gives the point , and going ends in the same point. This pair is valid.
and . ends in the point , while going the other way gives . This pair isn't commute.
and . . gives the point , while the other way gives . Not a valid pair
and . ends in the point , and also ends in . This pair works.
and . gives the point , and going the other way also ends in . This pair is valid.
Therefore, the answer is .
Note: It is easier to just visualize this problem instead of actually calculating points on paper.
~Tacos_are_yummy_1
Solution 3 (Specific)
Label the transformations as follows:
• a translation units to the right
• a -rotation counterclockwise about the origin
• a reflection across the -axis
• a dilation centered at the origin with scale factor
Now, we count each transformation individually. It is not hard to see that and are commutative (an easy way to test commutativity for some cases would be to have the original point on the -axis).
In total, transformation pairs commute.
~xHypotenuse
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.