# 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

~Interstigation

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 