document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210551>Question 1210551</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In equiangular octagaon EFGHIJKL, we know that EF = GH = IJ = KL = 1 and FG = HI = JK = LE = sqrt(2).  Find the area of the octagon. </SPAN>\n" );
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document.write( "Answer: <font color=red>7 square units</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "On a piece of graph paper, plot the points<BR>\n" );
document.write( "P = (0,0)<BR>\n" );
document.write( "Q = (3,0)<BR>\n" );
document.write( "R = (3,3)<BR>\n" );
document.write( "S = (0,3)<BR>\n" );
document.write( "This forms square PQRS which has area 3^2 = 9 square units.<BR>\n" );
document.write( "We'll use this value later.\r<BR>\n" );
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document.write( "Contained in this square will be octagon EFGHIJKL.<BR>\n" );
document.write( "We'll place the vertices at the following locations<BR>\n" );
document.write( "E = (1, 0)<BR>\n" );
document.write( "F = (2, 0)<BR>\n" );
document.write( "G = (3, 1)<BR>\n" );
document.write( "H = (3, 2)<BR>\n" );
document.write( "I = (2, 3)<BR>\n" );
document.write( "J = (1, 3)<BR>\n" );
document.write( "K = (0, 2)<BR>\n" );
document.write( "L = (0, 1)<BR>\n" );
document.write( "For horizontal or vertical segments, such as EF, you can simply count the number of spaces between the points to find the segment lengths.<BR>\n" );
document.write( "The distance formula could be used, but it would be overkill.<BR>\n" );
document.write( "You should find all horizontal and vertical segments are 1 unit each.\r<BR>\n" );
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document.write( "For diagonal pieces, use either the Distance Formula or the Pythagorean Theorem.<BR>\n" );
document.write( "Each diagonal piece is sqrt(2) units long.\r<BR>\n" );
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document.write( "You should find that each interior angle of the octagon is 135 degrees, which proves it to be equiangular. It's not regular since the sides aren't the same length. \r<BR>\n" );
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document.write( "Let's return to square PQRS.<BR>\n" );
document.write( "We need to clip a right triangle from each corner to form octagon EFGHIJKL.<BR>\n" );
document.write( "One such triangular piece to clip off would be triangle ELP, which has area =  base*height/2 = 1*1/2 = 0.5<BR>\n" );
document.write( "Four such congruent pieces to clip off would mean a total of 4*0.5 = 2 square units are removed.\r<BR>\n" );
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document.write( "Therefore octagon EFGHIJKL has area 9-2 = <font color=red>7 square units</font><BR>\n" );
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