document.write( "Question 1209520: (39) Square SQUR has sides of length x. If triangle SQE is equilateral, find the area of triangle QAU.
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Algebra.Com's Answer #851839 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Square SQUR has sides of length x. If triangle SQE is equilateral, find the area of triangle QAU.
\n" ); document.write( "Link to diagram: https://ibb.co/C58rZ09R
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\n" ); document.write( "\n" ); document.write( "        The solution in the post by @CPhill is  INCORRECT.\r
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document.write( "I will use coordinate plane (x,y).  To avoid missing coordinate x with the side length of the square, \r\n" );
document.write( "I will use 'a' for the square side length.\r\n" );
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document.write( "Let's place the origin of the coordinate system (x,y) at point S.\r\n" );
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document.write( "The line EQ has an equation\r\n" );
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document.write( "    y = \"sqrt%283%29%2Ax+%2B+b\"    (1)\r\n" );
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document.write( "since its slope is, obviously,  \"sqrt%283%29\".  To find 'b' in this equation, substitute coordinates of \r\n" );
document.write( "the vertex Q = {a,0) of the square into this equation\r\n" );
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document.write( "    0 = \"sqrt%283%29%2Aa+%2B+b\".\r\n" );
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document.write( "It gives  b = \"-sqrt%283%29%2Aa\".  So, equation of the line EQ is \r\n" );
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document.write( "    y = \"sqrt%283%29%2Ax+-+sqrt%283%29%2Aa%29\" = \"sqrt%283%29%2A%28x-a%29%29\".    (2)\r\n" );
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document.write( "We want to find the point A as the intersection of the line SU and the line EQ.\r\n" );
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document.write( "The line SU has the equation  y = -x;  so, we substitute y = -x into equation (2).  It gives\r\n" );
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document.write( "    -x = \"sqrt%283%29%2A%28x-a%29\" ,  \r\n" );
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document.write( "or  \r\n" );
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document.write( "    -x = \"sqrt%283%29%2Ax+-+sqrt%283%29%2Aa\",\r\n" );
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document.write( "    \"sqrt%283%29%2Aa\" = \"sqrt%283%29x+%2B+x\",\r\n" );
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document.write( "    x = \"%28sqrt%283%29%2Aa%29%2F%28sqrt%283%29%2B1%29\".\r\n" );
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document.write( "Thus we know now the x-coordinate of the point A.\r\n" );
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document.write( "So, now we can find the height h of the triangle QAU as the difference a-x:\r\n" );
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document.write( "    h = a-x = a - \"%28sqrt%283%29%2Aa%29%2F%28sqrt%283%29%2B1%29\" = \"%28sqrt%283%29%2Aa+%2B+a+-+sqrt%283%29%2Aa%29%2F%28sqrt%283%29%2B1%29\" = \"a%2F%28sqrt%283%29%2B1%29\".\r\n" );
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document.write( "You can rationalize the denominator\r\n" );
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document.write( "    h = \"%28a%2F%28sqrt%283%29%2B1%29%29%2A%28%28sqrt%283%29-1%29%2F%28sqrt%283%29-1%29%29\" = \"a%2A%28sqrt%283%29-1%29%2F%283-1%29\" = \"a%2A%28%28sqrt%283%29-1%29%2F2%29\".\r\n" );
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document.write( "Now the area of the triangle QAU is half the product of its base 'a' by its height h = \"a%2A%28%28sqrt%283%29-1%29%2F2%29\"\r\n" );
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document.write( "The  ANSWER  is :  the area of the triangle QAU is\r\n" );
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document.write( "    \"%281%2F2%29%2Aa%2Ah\" = \"%281%2F2%29%2Aa%2A%28a%2A%28sqrt%283%29-1%29%2F2%29%29\" = \"a%5E2%2A%28%28sqrt%283%29-1%29%2F4%29\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "                Regarding the post by @CPhill . . . \r
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\n" ); document.write( "\n" ); document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r
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\n" ); document.write( "\n" ); document.write( "The artificial intelligence is like a baby now. It is in the experimental stage
\n" ); document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r
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\n" ); document.write( "\n" ); document.write( "                It has no feeling of shame - it is shameless.\r
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\n" ); document.write( "\n" ); document.write( "This time, again,  it made an error.\r
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\n" ); document.write( "\n" ); document.write( "Although the @CPhill' solution are copy-paste  Google  AI solutions,  there is one essential difference.\r
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\n" ); document.write( "\n" ); document.write( "Every time,  Google  AI  makes a note at the end of its solutions that  Google  AI  is experimental
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\n" ); document.write( "\n" ); document.write( "All @CPhill' solutions are copy-paste of  Google  AI  solutions, with one difference:
\n" ); document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.
\n" ); document.write( "So,  he  NEVER  SAYS  TRUTH.\r
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\n" ); document.write( "\n" ); document.write( "Every time,  @CPhill embarrassed to tell the truth.\r
\n" ); document.write( "\n" ); document.write( "But I am not embarrassing to tell the truth,  as it is my duty at this forum.\r
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\n" ); document.write( "\n" ); document.write( "And the last my comment.\r
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\n" ); document.write( "\n" ); document.write( "When you obtain such posts from @CPhill,  remember,  that  NOBODY  is responsible for their correctness,
\n" ); document.write( "until the specialists and experts will check and confirm their correctness.\r
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\n" ); document.write( "\n" ); document.write( "Without it,  their reliability is  ZERO and their creadability is  ZERO,  too.\r
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