2015 AMC 8 Problems/Problem 2: Difference between revisions

Revision as of 14:24, 25 November 2015

Point $O$ is the center of the regular octagon $ABCDEFGH$ , and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?

$\textbf{(A) }\frac{11}{32} \quad\textbf{(B) }\frac{3}{8} \quad\textbf{(C) }\frac{13}{32} \quad\textbf{(D) }\frac{7}{16}\quad \textbf{(E) }\frac{15}{32}$

[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(" $A$ ",A,dir(45)); dot(" $B$ ",B,dir(90)); dot(" $C$ ",C,dir(135)); dot(" $D$ ",D,dir(180)); dot(" $E$ ",E,dir(-135)); dot(" $F$ ",F,dir(-90)); dot(" $G$ ",G,dir(-45)); dot(" $H$ ",H,dir(0)); dot(" $X$ ",X,dir(135/2)); dot(" $O$ ",O,dir(0)); draw(E--O--X); [/asy]

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2015 AMC 8 Problems/Problem 2: Difference between revisions

Revision as of 14:24, 25 November 2015

@@ Line 2: / Line 2: @@
 <math>\textbf{(A) }\frac{11}{32} \quad\textbf{(B) }\frac{3}{8} \quad\textbf{(C) }\frac{13}{32} \quad\textbf{(D) }\frac{7}{16}\quad \textbf{(E) }\frac{15}{32}</math>
+[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("<math>A</math>",A,dir(45));
+dot("<math>B</math>",B,dir(90));
+dot("<math>C</math>",C,dir(135));
+dot("<math>D</math>",D,dir(180));
+dot("<math>E</math>",E,dir(-135));
+dot("<math>F</math>",F,dir(-90));
+dot("<math>G</math>",G,dir(-45));
+dot("<math>H</math>",H,dir(0));
+dot("<math>X</math>",X,dir(135/2));
+dot("<math>O</math>",O,dir(0));
+draw(E--O--X);
+[/asy]