MathDB
Shaded Octagon

Source: 2015 AMC 8 #2

November 25, 2015
AMC 8geometry

Problem Statement

Point OO is the center of the regular octagon ABCDEFGHABCDEFGH, and XX is the midpoint of the side AB.\overline{AB}. What fraction of the area of the octagon is shaded?
<spanclass=latexbold>(A)</span>1132<spanclass=latexbold>(B)</span>38<spanclass=latexbold>(C)</span>1332<spanclass=latexbold>(D)</span>716<spanclass=latexbold>(E)</span>1532<span class='latex-bold'>(A) </span>\frac{11}{32} \qquad<span class='latex-bold'>(B) </span>\frac{3}{8} \qquad<span class='latex-bold'>(C) </span>\frac{13}{32} \qquad<span class='latex-bold'>(D) </span>\frac{7}{16}\qquad <span class='latex-bold'>(E) </span>\frac{15}{32}
[asy] pair A,B,C,D,E,F,G,H,O,X; A=dir(45); B=dir(90); C=dir(135); D=dir(180); E=dir(-135); F=dir(-90); G=dir(-45); H=dir(0); O=(0,0); X=midpoint(A--B);
fill(X--B--C--D--E--O--cycle,rgb(0.75,0.75,0.75)); draw(A--B--C--D--E--F--G--H--cycle);
dot("AA",A,dir(45)); dot("BB",B,dir(90)); dot("CC",C,dir(135)); dot("DD",D,dir(180)); dot("EE",E,dir(-135)); dot("FF",F,dir(-90)); dot("GG",G,dir(-45)); dot("HH",H,dir(0)); dot("XX",X,dir(135/2)); dot("OO",O,dir(0)); draw(E--O--X); [/asy]