document.write( "Question 1209482: Vertices of a regular octagon are connected as shown. What fraction of the octagon is shaded?\r
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Algebra.Com's Answer #848933 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "Vertices of a regular octagon are connected as shown. What fraction of the octagon $\"highlight%28area%29\"$ is shaded?\r
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document.write( "Let \"a\" be the side length of this regular octagon.\r\n" );
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document.write( "Then the area of the whole octagon is (3a)^2 - 2a^2 = 9a^2 - 2a^2 = 7a^2.\r\n" );
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document.write( "It is because the area of the square accommodating the octagon is (3a)^2,\r\n" );
document.write( "from which we should subtract/(cut) four halves of squares with side length \"a\".\r\n" );
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document.write( "OK.  Now, in the picture, we have the shadowed triangle and another, greater triangle,\r\n" );
document.write( "that accommodates the shadowed triangle.\r\n" );
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document.write( "This greater triangle area is, obviously,  , since this greater triangle is half \r\n" );
document.write( "of the rectangle with sides \"a\" and 3a.\r\n" );
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document.write( "Now our goal is to estimate the area of the shadowed triangle.\r\n" );
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document.write( "These two triangles, the shadowed and the greater one, are right-angled triangles with the common\r\n" );
document.write( "acute angle. So, these two triangles are similar.\r\n" );
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document.write( "To estimate the area of the shadowed triangle, it is enough to find the coefficient of similarity \r\n" );
document.write( "of the triangles.\r\n" );
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document.write( "Compare the longest legs of these two triangles.\r\n" );
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document.write( "For the greater  triangle, the length of its longest leg is 3a.\r\n" );
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document.write( "For the shadowed triangle, the length of its longest leg is the radius of the circumscribed circle \r\n" );
document.write( "about the regular octagon.\r\n" );
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document.write( "The formula expressing the radius of the circumscribed circle via the side of the regular octagon is\r\n" );
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document.write( "    r = .\r\n" );
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document.write( "So, the similarity coefficient for the two triangles is (as the ratio of the smaller to the larger)\r\n" );
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document.write( "    k =  =  = .\r\n" );
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document.write( "Well known fact is that sin(22.5°) = .\r\n" );
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document.write( "Therefore, the similarity coefficient for the two triangles is  \r\n" );
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document.write( "    k =  = .\r\n" );
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document.write( "Thus the area of the shadowed triangle is\r\n" );
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document.write( "    area =  =  = .\r\n" );
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document.write( "To calculate the fraction of the area, we should relate it to 7a^2, the area of the regular octagon.\r\n" );
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document.write( "Thus the  ANSWER  is:  fraction =  = 0.0406454  (approximately).\r\n" );
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