document.write( "Question 1135915: from the point inside a square the distance to three corners are 4,5 and 6 m respectively. Find the length of the side of a square. \n" ); document.write( "

Algebra.Com's Answer #753645 by ikleyn(52802) $\"\"$ $\"About$

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document.write( "Let ABCD be the square with the side of the length \"a\" in a coordinate plane,\r\n" );
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document.write( "    A = (0,0),  B = (a,0),  C = (a,a)  and  D = (0,a).\r\n" );
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document.write( "Let (x,y) be the point inside the square ABCD with the distance 4 from A, 5 from D  and  6 from B.\r\n" );
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document.write( "Thus we have these three equations (\"distances\")\r\n" );
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document.write( "    x^2     + y^2     = 4^2,      (1)\r\n" );
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document.write( "    (a-x)^2 + y^2     = 6^2,      (2)\r\n" );
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document.write( "    x^2     + (y-a)^2 = 5^2.      (3)\r\n" );
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document.write( "Making FOIL in equations (2) and (3), I can re-write them in this form\r\n" );
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document.write( "    x^2             + y^2 = 16,   (4)    (= same as (1) )\r\n" );
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document.write( "    a^2 - 2ax + x^2 + y^2 = 36,   (5)\r\n" );
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document.write( "    x^2 + y^2 - 2ay + a^2 = 25.   (6)\r\n" );
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document.write( "Replacing  x^2 + y^2 by 16  in equations (5) and (6), I obtain new equations instead of them\r\n" );
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document.write( "    a^2 - 2ax = 20                (7)\r\n" );
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document.write( "    a^2 - 2ay =  9                (8)\r\n" );
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document.write( "From equations (7) and (8),  x = ,  y = .\r\n" );
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document.write( "Substituting these expressions for x and y into equation (4), you get\r\n" );
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document.write( "     +  = ,\r\n" );
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document.write( "or, simplifying\r\n" );
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document.write( "     +  = ,\r\n" );
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document.write( "     = 0.\r\n" );
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document.write( "From this bi-quadratic equation, you get for , by applying the quadratic formula\r\n" );
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document.write( "     =  = .\r\n" );
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document.write( "The smaller value does not work for \"a\" (as it is easy to check), leaving the larger value\r\n" );
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document.write( "     = \r\n" );
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document.write( "as the only meaningful.\r\n" );
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document.write( "Thus  a =  = 7.534 (approximately).\r\n" );
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