document.write( "Question 1143644: A point O is inside a square lot. If the distances from point O to the three successive corners of the square lot are 5m, 3m, 4m respectively. Determine the area of the square lot.\r
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Algebra.Com's Answer #764504 by ikleyn(52858) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            Take a page of paper and a pen or a pencil and plot it on your own.\r
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\n" ); document.write( "\n" ); document.write( "            We are not an art studio to make plots for you.\r
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\n" ); document.write( "\n" ); document.write( "            Instead of plotting the figure, I will solve the problem, from the beginning to the end.   OK ?\r
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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 3 from A, 4 from D  and  5 from B.\r\n" );
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document.write( "Then we have these three equations (\"distances\")\r\n" );
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document.write( "    x^2     + y^2     = 3^2,      (1)\r\n" );
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document.write( "    (a-x)^2 + y^2     = 5^2,      (2)\r\n" );
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document.write( "    x^2     + (y-a)^2 = 4^2.      (3)\r\n" );
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document.write( "Making FOIL in equations (2) and (3), I can re-write (1), (2) and (3) in this form\r\n" );
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document.write( "    x^2             + y^2 =  9,   (4)    (= same as (1) )\r\n" );
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document.write( "    a^2 - 2ax + x^2 + y^2 = 25,   (5)\r\n" );
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document.write( "    x^2 + y^2 - 2ay + a^2 = 16.   (6)\r\n" );
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document.write( "Replacing  x^2 + y^2 by 9  in equations (5) and (6), I obtain new equations instead of them\r\n" );
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document.write( "    a^2 - 2ax = 16                (7)\r\n" );
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document.write( "    a^2 - 2ay =  7                (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( "     +  = {{36a^2}}},\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  the area of the square is   =  = 36.863 square units (approximately).\r\n" );
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\n" ); document.write( "\n" ); document.write( "It is not for the first time such problem comes to the forum.\r
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\n" ); document.write( "\n" ); document.write( "Some time ago I solved similar problem here under this link\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Parallelograms/Parallelograms.faq.question.1135915.html\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Parallelograms/Parallelograms.faq.question.1135915.html\r
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\n" ); document.write( "\n" ); document.write( "Thanks for posting this interesting problem.\r
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\n" ); document.write( "\n" ); document.write( "It was a pleasure for me to solve it again (even for the second time), because the solution is nice.\r
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