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Difference between revisions of "2022 AMC 8 Problems/Problem 4"

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size(3cm);
 
size(3cm);
 
draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(0.6,0.25));
 
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));
+
draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1));  
 
</asy>
 
</asy>
 
When <b>M</b> is then reflected over the line <math>p,</math> we obtain the following diagram:
 
When <b>M</b> is then reflected over the line <math>p,</math> we obtain the following diagram:
Line 48: Line 48:
 
size(3cm);
 
size(3cm);
 
draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(0.6,-0.25));
 
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));
+
draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1));  
 
</asy>
 
</asy>
 
Therefore, the answer is <math>\boxed{\textbf{(E)}}.</math>
 
Therefore, the answer is <math>\boxed{\textbf{(E)}}.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM
 +
 +
==Video Solution (A Clever Explanation You’ll Get Instantly)==
 +
https://youtu.be/tYWp6fcUAik?si=V8hv_zOn_zYOi9E5&t=350
 +
~hsnacademy
 +
 +
==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}}
 +
 +
[[Category:Introductory Geometry Problems]]

Latest revision as of 16:48, 4 June 2025

Problem

The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$. 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

When M is first reflected over the line $q,$ 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 M is then reflected over the line $p,$ 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 $\boxed{\textbf{(E)}}.$

~MRENTHUSIASM

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/tYWp6fcUAik?si=V8hv_zOn_zYOi9E5&t=350 ~hsnacademy

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

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

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