MathDB
Reflection over 2 Lines (2022 AMC 8 P4)

Source:

January 26, 2022
geometrygeometric transformationreflectionAMC 82022 AMC 8

Problem Statement

The letter M in the figure below is first reflected over the line qq and then reflected over the line pp. What is the resulting image? [asy] // pog diagram usepackage("newtxtext"); size(3cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label("<spanclass=latexbold>M</span><span class='latex-bold'>\textsf{M</span>}",(0.25,0.6)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("pp", (-1,0),NE); label("qq", (-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)*"<spanclass=latexbold>M</span><span class='latex-bold'>\textsf{M</span>}",(0.6,-0.25)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("<spanclass=latexbold>(A)</span><span class='latex-bold'>(A)</span>",(-1,1),W); draw((2,0)--(4,0)); draw((3,-1)--(3,1)); label(rotate(270)*"<spanclass=latexbold>M</span><span class='latex-bold'>\textsf{M</span>}",(2.8,0.7)); draw((2.2,-0.8)--(3.8,0.8),linewidth(1.1)); label("<spanclass=latexbold>(B)</span><span class='latex-bold'>(B)</span>",(2,1),W); draw((5,0)--(7,0)); draw((6,-1)--(6,1)); label(rotate(90)*"<spanclass=latexbold>M</span><span class='latex-bold'>\textsf{M</span>}",(5.4,0.2)); draw((5.2,-0.8)--(6.8,0.8),linewidth(1.1)); label("<spanclass=latexbold>(C)</span><span class='latex-bold'>(C)</span>",(5,1),W); draw((-1,-2.5)--(1,-2.5)); draw((0,-3.5)--(0,-1.5)); label(rotate(180)*"<spanclass=latexbold>M</span><span class='latex-bold'>\textsf{M</span>}",(-0.25,-3.1)); draw((-0.8,-3.3)--(0.8,-1.7),linewidth(1.1)); label("<spanclass=latexbold>(D)</span><span class='latex-bold'>(D)</span>",(-1,-1.5),W); draw((2,-2.5)--(4,-2.5)); draw((3,-3.5)--(3,-1.5)); label(rotate(270)*"<spanclass=latexbold>M</span><span class='latex-bold'>\textsf{M</span>}",(3.6,-2.75)); draw((2.2,-3.3)--(3.8,-1.7),linewidth(1.1)); label("<spanclass=latexbold>(E)</span><span class='latex-bold'>(E)</span>",(2,-1.5),W); [/asy]