Difference between revisions of "2024 AMC 10A Problems/Problem 13"
Xhypotenuse (talk | contribs) |
MRENTHUSIASM (talk | contribs) (→Solution 3 (Specific)) |
||
(15 intermediate revisions by 6 users not shown) | |||
Line 1: | Line 1: | ||
== Problem == | == 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: | + | 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 <math>2</math> units to the right, | |
− | + | * a <math>90^{\circ}</math>-rotation counterclockwise about the origin, | |
− | + | * a reflection across the <math>x</math>-axis, and | |
− | + | * a dilation centered at the origin with scale factor <math>2.</math> | |
− | Of the 6 pairs of distinct transformations from this list, how many commute? | + | Of the <math>6</math> 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> | ||
− | <math> | + | == 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: | ||
− | == Solution | + | * <math>T_1: \ (x,y)\to(x+2,y)</math> |
+ | |||
+ | * <math>T_2: \ (x,y)\to(-y,x)</math> | ||
+ | |||
+ | * <math>T_3: \ (x,y)\to(x,-y)</math> | ||
+ | |||
+ | * <math>T_4: \ (x,y)\to(2x,2y)</math> | ||
+ | |||
+ | Note that: | ||
+ | <ul style="list-style-type:square;"> | ||
+ | <li>Applying <math>T_1</math> and then <math>T_2</math> gives <math>(x,y)\to(x+2,y)\to(-y,x+2).</math> <p> | ||
+ | Applying <math>T_2</math> and then <math>T_1</math> gives <math>(x,y)\to(-y,x)\to(-y+2,x).</math> <p> | ||
+ | Therefore, <math>T_1</math> and <math>T_2</math> do not commute. One counterexample is the preimage <math>(0,0).</math> | ||
+ | </li><p> | ||
+ | <li>Applying <math>T_1</math> and then <math>T_3</math> gives <math>(x,y)\to(x+2,y)\to(x+2,-y).</math> <p> | ||
+ | Applying <math>T_3</math> and then <math>T_1</math> gives <math>(x,y)\to(x,-y)\to(x+2,-y).</math> <p> | ||
+ | Therefore, <math>T_1</math> and <math>T_3</math> commute. They form a <b>glide reflection</b>. | ||
+ | </li><p> | ||
+ | <li>Applying <math>T_1</math> and then <math>T_4</math> gives <math>(x,y)\to(x+2,y)\to(2x+4,2y).</math> <p> | ||
+ | Applying <math>T_4</math> and then <math>T_1</math> gives <math>(x,y)\to(2x,2y)\to(2x+2,2y).</math> <p> | ||
+ | Therefore, <math>T_1</math> and <math>T_4</math> do not commute. One counterexample is the preimage <math>(0,0).</math> | ||
+ | </li><p> | ||
+ | <li>Applying <math>T_2</math> and then <math>T_3</math> gives <math>(x,y)\to(-y,x)\to(-y,-x).</math> <p> | ||
+ | Applying <math>T_3</math> and then <math>T_2</math> gives <math>(x,y)\to(x,-y)\to(y,x).</math> <p> | ||
+ | Therefore, <math>T_2</math> and <math>T_3</math> do not commute. One counterexample is the preimage <math>(1,0).</math> | ||
+ | </li><p> | ||
+ | <li>Applying <math>T_2</math> and then <math>T_4</math> gives <math>(x,y)\to(-y,x)\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. | ||
+ | </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> | ||
+ | </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> | ||
+ | |||
+ | <u><b>Remark</b></u> | ||
+ | |||
+ | To show that two transformations do not commute, we only need one counterexample. | ||
+ | |||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | == 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> | ||
<math>1.</math> <math>W</math> and <math>X</math>. <math>W\rightarrow X</math> ends with the point <math>(0,3)</math>. Going <math>X\rightarrow W</math> ends in the point <math>(1,2)</math>, so this pair does not work | <math>1.</math> <math>W</math> and <math>X</math>. <math>W\rightarrow X</math> ends with the point <math>(0,3)</math>. Going <math>X\rightarrow W</math> ends in the point <math>(1,2)</math>, so this pair does not work | ||
Line 41: | Line 87: | ||
<math>6.</math> <math>Y</math> and <math>Z</math>. <math>Y\rightarrow Z</math> gives the point <math>(2,0)</math>, and going the other way also ends in <math>(2,0)</math>. This pair is valid. | <math>6.</math> <math>Y</math> and <math>Z</math>. <math>Y\rightarrow Z</math> gives the point <math>(2,0)</math>, and going the other way also ends in <math>(2,0)</math>. This pair is valid. | ||
− | Therefore, the answer is <math>\boxed{\ | + | Therefore, the answer is <math>\boxed{\textbf{(C) }3}</math>. |
Note: It is easier to just visualize this problem instead of actually calculating points on paper. | Note: It is easier to just visualize this problem instead of actually calculating points on paper. | ||
+ | ~Tacos_are_yummy_1 | ||
− | == Solution | + | == Solution 3 (Specific) == |
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 <math>90^{\circ}</math>-rotation counterclockwise about the origin <math>(B)</math> | ||
+ | |||
+ | • a reflection across the <math>x</math>-axis <math>(C)</math> | ||
− | • a | + | • 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). | |
− | + | In total, <math>\boxed{\textbf{(C) }3}</math> transformation pairs commute. | |
+ | ~xHypotenuse | ||
− | + | ==Solution 4 (Quick With Reasoning)== | |
+ | Alright so realize that obviously rotation and dilation are commute, if you ever looked at the polar plane. Also, reflecting around the x-axis means that moving to the right two units and dilating are commute. You can easily see this by taking an arbitrary point and applying the transformations. The rest don’t work since you can easily test out using random points on the plane, which means the answer is <math>\boxed{\textbf{(C) }3}</math>. | ||
− | + | ~EaZ_Shadow | |
+ | == Video Solution by Pi Academy == | ||
− | + | https://youtu.be/ABkKz0gS1MU?si=ZQBgDMRaJmMPSSMM | |
+ | |||
+ | ==Video Solution 1 by Power Solve == | ||
+ | https://youtu.be/QVDbm5sDxxU | ||
+ | |||
+ | ==Video Solution 2 by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=6SQ74nt3ynw | ||
==See also== | ==See also== | ||
{{AMC10 box|year=2024|ab=A|num-b=12|num-a=14}} | {{AMC10 box|year=2024|ab=A|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 11:31, 18 December 2024
Contents
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. One counterexample is the preimage
- 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. One counterexample is the preimage
- Applying and then gives
Applying and then gives
Therefore, and do not commute. One counterexample is the preimage
- 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
Remark
To show that two transformations do not commute, we only need one counterexample.
~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
Solution 4 (Quick With Reasoning)
Alright so realize that obviously rotation and dilation are commute, if you ever looked at the polar plane. Also, reflecting around the x-axis means that moving to the right two units and dilating are commute. You can easily see this by taking an arbitrary point and applying the transformations. The rest don’t work since you can easily test out using random points on the plane, which means the answer is .
~EaZ_Shadow
Video Solution by Pi Academy
https://youtu.be/ABkKz0gS1MU?si=ZQBgDMRaJmMPSSMM
Video Solution 1 by Power Solve
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=6SQ74nt3ynw
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.