Algebra.Com's Answer #811166 by ikleyn(52813)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "The diagonals of right trapezoid are perpendicular.
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document.write( "What is the area of the trapezoid if the parallel bases
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document.write( "measure 8 cm and 18 cm?
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document.write( " It is really first class geometry problem, \r
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document.write( " and I will show you absolutely unexpected solution, right now.\r
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document.write( "Make a sketch.\r\n" );
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document.write( "Let d be the shorter diagonal and D be the longer diagonal of the trapezoid.\r\n" );
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document.write( "Let H be the altitude of the trapezoid.\r\n" );
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document.write( "Then from Pythagoras, you have these two equations\r\n" );
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document.write( " 8^2 + H^2 = d^2 (1)\r\n" );
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document.write( " 18^2 + H^2 = D^2 (2)\r\n" );
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document.write( "Since the diagonals are perpendicular, there is this formula for the trapezoid area A\r\n" );
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document.write( " A =  . (3)\r\n" );
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document.write( "From the other side, there is a classic formula for the trapezoid area as \r\n" );
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document.write( " A =  (4)\r\n" );
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document.write( "Equating (3) and (4), you have\r\n" );
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document.write( " = 13H, \r\n" );
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document.write( "or\r\n" );
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document.write( " d*D = 26H. (5)\r\n" );
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document.write( "Thus we have three equations (1), (2) and (5) for three unknowns d, D and H.\r\n" );
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document.write( "Now the most interesting part of the solution is starting.\r\n" );
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document.write( "Multiply equations (1) and (2). You will get\r\n" );
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document.write( " (64 + H^2)*(324 + H^2) = (d*D)^2. (6)\r\n" );
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document.write( "Replace (d*D)^2 in the right side of the equation (6) by (26*H)^2, based on (5).\r\n" );
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document.write( "By doing this way, you reduce the system of three equations in three unknowns (1), (2) and (5) \r\n" );
document.write( "to one single equation for H^2\r\n" );
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document.write( " (64 + H^2)*(324 + H^2) = 676*H^2.\r\n" );
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document.write( "Simplify and solve (it is biquadratic equation for H)\r\n" );
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document.write( " 64*324 + 324H^2 + 64H^2 + H^4 = 676H^2\r\n" );
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document.write( " H^4 - 288H^2 + 20736 = 0\r\n" );
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document.write( "Factor left side\r\n" );
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document.write( " (H^2-144)^2 = 0.\r\n" );
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document.write( "It gives H^2 = 144; hence, H =  = 12.\r\n" );
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document.write( "Thus we found that the altitude of the trapezoid H is 12 cm.\r\n" );
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document.write( "Now the area of the trapezoid is  = 13*12 = 156 square centimeters. ANSWER\r\n" );
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document.write( "The problem is solved and all necessary explanations are given.\r
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