2022 AMC 8 Problems/Problem 4

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

https://youtu.be/Ij9pAy6tQSg?t=259

~Interstigation

See Also

2022 AMC 8 (ProblemsAnswer KeyResources)
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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