Difference between revisions of "2022 AMC 8 Problems/Problem 4"
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==Problem== | ==Problem== | ||
| − | + | ||
| + | The letter <b>M</b> in the figure below is first reflected over the line <math>q</math> and then reflected over the line <math>p</math>. What is the resulting image? | ||
| + | |||
| + | <asy> | ||
| + | // pog diagram | ||
| + | usepackage("newtxtext"); | ||
| + | size(3cm); | ||
| + | draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label("$\textbf{\textsf{M}}$",(0.25,0.6)); | ||
| + | draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$p$", (-1,0),NE); label("$q$", (-0.75,-0.75), N*1.5); | ||
| + | </asy> | ||
| + | |||
| + | <asy> | ||
| + | // pog diagram | ||
| + | usepackage("newtxtext"); | ||
| + | size(12.5cm); | ||
| + | draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(0.6,-0.25)); | ||
| + | draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); | ||
| + | label("$\textbf{(A)}$",(-1,1),W); | ||
| + | draw((2,0)--(4,0)); draw((3,-1)--(3,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(2.8,0.7)); | ||
| + | draw((2.2,-0.8)--(3.8,0.8),linewidth(1.1)); | ||
| + | label("$\textbf{(B)}$",(2,1),W); | ||
| + | draw((5,0)--(7,0)); draw((6,-1)--(6,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(5.4,0.2)); | ||
| + | draw((5.2,-0.8)--(6.8,0.8),linewidth(1.1)); | ||
| + | label("$\textbf{(C)}$",(5,1),W); | ||
| + | draw((-1,-2.5)--(1,-2.5)); draw((0,-3.5)--(0,-1.5)); label(rotate(180)*"$\textbf{\textsf{M}}$",(-0.25,-3.1)); | ||
| + | draw((-0.8,-3.3)--(0.8,-1.7),linewidth(1.1)); | ||
| + | label("$\textbf{(D)}$",(-1,-1.5),W); | ||
| + | draw((2,-2.5)--(4,-2.5)); draw((3,-3.5)--(3,-1.5)); label(rotate(270)*"$\textbf{\textsf{M}}$",(3.6,-2.75)); | ||
| + | draw((2.2,-3.3)--(3.8,-1.7),linewidth(1.1)); | ||
| + | label("$\textbf{(E)}$",(2,-1.5),W); | ||
| + | </asy> | ||
==Solution== | ==Solution== | ||
| + | |||
| + | When <b>M</b> is first reflected over the line <math>q,</math> we obtain the following diagram: | ||
| + | <asy> | ||
| + | /* Made by MRENTHUSIASM */ | ||
| + | usepackage("newtxtext"); | ||
| + | size(3cm); | ||
| + | draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(0.6,0.25)); | ||
| + | draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); | ||
| + | </asy> | ||
| + | When <b>M</b> is then reflected over the line <math>p,</math> we obtain the following diagram: | ||
| + | <asy> | ||
| + | /* Made by MRENTHUSIASM */ | ||
| + | usepackage("newtxtext"); | ||
| + | size(3cm); | ||
| + | draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(0.6,-0.25)); | ||
| + | draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); | ||
| + | </asy> | ||
| + | Therefore, the answer is <math>\boxed{\textbf{(E)}}.</math> | ||
| + | |||
| + | ~MRENTHUSIASM | ||
| + | |||
| + | ==Video Solution 1 by Math-X (First understand the problem!!!)== | ||
| + | https://youtu.be/oUEa7AjMF2A?si=l-XMI73aofM6q4ye&t=320 | ||
| + | |||
| + | ~Math-X | ||
| + | |||
| + | ==Video Solution 2 (CREATIVE THINKING!!!)== | ||
| + | https://youtu.be/lItyKxX_qB8 | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution 3== | ||
| + | https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=259 | ||
| + | |||
| + | ~Interstigation | ||
| + | |||
| + | ==Video Solution 4== | ||
| + | https://youtu.be/c-D-bj2htGE | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==Video Solution 5== | ||
| + | https://youtu.be/Q0R6dnIO95Y?t=182 | ||
| + | |||
| + | ~STEMbreezy | ||
| + | |||
| + | ==Video Solution 6== | ||
| + | https://www.youtube.com/watch?v=6kkJVsOKcuk | ||
| + | |||
| + | ~harungurcan | ||
| + | |||
| + | ==Video Solution 7 by Dr. David== | ||
| + | |||
| + | https://youtu.be/hh3DbQ9Rd5Y | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2022|num-b=3|num-a=5}} | {{AMC8 box|year=2022|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 01:48, 19 November 2024
Contents
Problem
The letter M in the figure below is first reflected over the line 
and then reflected over the line 
. What is the resulting image?
![[asy] // pog diagram usepackage("newtxtext"); size(3cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label("$\textbf{\textsf{M}}$",(0.25,0.6)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$p$", (-1,0),NE); label("$q$", (-0.75,-0.75), N*1.5); [/asy]](http://latex.artofproblemsolving.com/c/3/8/c38b67d113d5fbf1b6d6eae0d87161b539dbd849.png)
![[asy] // pog diagram usepackage("newtxtext"); size(12.5cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(0.6,-0.25)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$\textbf{(A)}$",(-1,1),W); draw((2,0)--(4,0)); draw((3,-1)--(3,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(2.8,0.7)); draw((2.2,-0.8)--(3.8,0.8),linewidth(1.1)); label("$\textbf{(B)}$",(2,1),W); draw((5,0)--(7,0)); draw((6,-1)--(6,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(5.4,0.2)); draw((5.2,-0.8)--(6.8,0.8),linewidth(1.1)); label("$\textbf{(C)}$",(5,1),W); draw((-1,-2.5)--(1,-2.5)); draw((0,-3.5)--(0,-1.5)); label(rotate(180)*"$\textbf{\textsf{M}}$",(-0.25,-3.1)); draw((-0.8,-3.3)--(0.8,-1.7),linewidth(1.1)); label("$\textbf{(D)}$",(-1,-1.5),W); draw((2,-2.5)--(4,-2.5)); draw((3,-3.5)--(3,-1.5)); label(rotate(270)*"$\textbf{\textsf{M}}$",(3.6,-2.75)); draw((2.2,-3.3)--(3.8,-1.7),linewidth(1.1)); label("$\textbf{(E)}$",(2,-1.5),W); [/asy]](http://latex.artofproblemsolving.com/1/9/b/19b58c8f6aafe00e0ad5454583fda04c3567f944.png)
Solution
When M is first reflected over the line 
we obtain the following diagram:
![[asy] /* Made by MRENTHUSIASM */ usepackage("newtxtext"); size(3cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(0.6,0.25)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); [/asy]](http://latex.artofproblemsolving.com/4/0/c/40c26208a133e8893a6c0dffc050e5c046953f21.png)
When M is then reflected over the line 
we obtain the following diagram:
![[asy] /* Made by MRENTHUSIASM */ usepackage("newtxtext"); size(3cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(0.6,-0.25)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); [/asy]](http://latex.artofproblemsolving.com/d/5/f/d5fb7ca1ab4567bbf6f1786661b033a767d9756c.png)
Therefore, the answer is 
~MRENTHUSIASM
Video Solution 1 by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=l-XMI73aofM6q4ye&t=320
~Math-X
Video Solution 2 (CREATIVE THINKING!!!)
~Education, the Study of Everything
Video Solution 3
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=259
~Interstigation
Video Solution 4
~savannahsolver
Video Solution 5
https://youtu.be/Q0R6dnIO95Y?t=182
~STEMbreezy
Video Solution 6
https://www.youtube.com/watch?v=6kkJVsOKcuk
~harungurcan
Video Solution 7 by Dr. David
See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
