Algebra.Com's Answer #853645 by ikleyn(53646)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "In equiangular octagon EFGHIJKL, we know that EF = GH = IJ = KL = 1 and FG = HI = JK = LE = sqrt(2).
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document.write( "Find the area of the octagon.
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document.write( "Let's write the sequence of side lengths in the row\r\n" );
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document.write( " EF FG GH HI IJ JK KL LE\r\n" );
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document.write( " 1 sqrt(2) 1 sqrt(2) 1 sqrt(2) 1 sqrt(2) <<<---=== (1)\r\n" );
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document.write( "You see the repeating pattern as a cycle.\r\n" );
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document.write( "All interior angles are  =  =  =  = 45*3 = 135 degrees.\r\n" );
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document.write( "You can calculate the length of the diagonal EG using the cosine law formula\r\n" );
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document.write( "  =  =  =  = 5.\r\n" );
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document.write( "so EG =  .\r\n" );
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document.write( "Obviously, all such sides EG, GI, IK and KE have the same length due to the same reason (the same logic).\r\n" );
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document.write( "Next, this octagon has a remarkable symmetry: if you rotate it in a way that vertex E goes to vertex G\r\n" );
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document.write( " E ---> G,\r\n" );
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document.write( "then the new octagon (the image under this rotation) will coincide with the original octagon.\r\n" );
document.write( "(simply because the sequence of side lengths (1) will be the same and all interior angles are congruent).\r\n" );
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document.write( "It means that the quadrilateral FHJK will map into and onto itself.\r\n" );
document.write( "It means that this quadrilateral is a square.\r\n" );
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document.write( " If such reasoning confuses you, you can notice that sides FH and HJ of the quadrilateral FHJK are orthogonal\r\n" );
document.write( " since two times angle 135 degs is 270 degs. And, similarly, quadrilateral FHJK has all his consecutive sides \r\n" );
document.write( " orthogonal with equal lengths, so this quadrilateral is a square.\r\n" );
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document.write( "Now, the area of the square FHJK is the square of its side, i.e.  = 5.\r\n" );
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document.write( "Now you can calculate the area of triangle EFG\r\n" );
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document.write( "  =  =  =  =  .\r\n" );
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document.write( "We have 4 such triangles as EFG, so their total area is  = 2.\r\n" );
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document.write( "Now the total area of the octagon EFGHIJKL is the sum of the area of square FHGK PLUS four triangles\r\n" );
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document.write( " 5 + 2 = 7.\r\n" );
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document.write( "At this point, the solution is complete.\r\n" );
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document.write( "The area of the octagon EFGHIJKL is 7 square units. ANSWER\r\n" );
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document.write( "Solved.\r
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document.write( "Nice problem. It is a fun to solve it.\r
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