Difference between revisions of "2015 AMC 8 Problems/Problem 2"
| 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> | <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] | ||
Revision as of 14:24, 25 November 2015
Point 
is the center of the regular octagon 
, and 
is the midpoint of the side 
What fraction of the area of the octagon is shaded?
[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,dir(45));
dot("
",B,dir(90));
dot("
",C,dir(135));
dot("
",D,dir(180));
dot("
",E,dir(-135));
dot("
",F,dir(-90));
dot("
",G,dir(-45));
dot("
",H,dir(0));
dot("",X,dir(135/2));
dot("",O,dir(0));
draw(E--O--X);
[/asy]
