Algebra.Com's Answer #852836 by ikleyn(53299)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "In rectangle ABCD, corner A is folded over crease DE to point F on BC. Find BC.
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "Make a sketch following my descriptions/instructions.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We consider rectangle ABCD. For simplicity, assume that the rectangle is \"vertical\",\r\n" );
document.write( "i.e. its vertical dimension 'b' = AD = BC is greater than (or equal to) its horizontal dimension 'a' = AB = CD.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We fold corner A, so point A comes to point F on BC: so, point F is the image of point A\r\n" );
document.write( "under folding, and F is on BC between points B and C. <<<---=== It is in accordance with the given part.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then it is clear that at this folding the side AB transforms to interval BF;\r\n" );
document.write( "so, AB = BF = a.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is also clear that at this folding the side AD becomes FD; therefore, DF = b.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Consider right-angled triangle FCD. For its leg FC we can write\r\n" );
document.write( "\r\n" );
document.write( " FC =  =  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now interval BC is the sum of two intervals\r\n" );
document.write( "\r\n" );
document.write( " BC = BF + FC = a +  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "But BC = b; so, we can rewrite the last equation in the form\r\n" );
document.write( "\r\n" );
document.write( " b = a + . (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we will perform formal transformations with this equation. From (1) we have, moving 'a' on the left side\r\n" );
document.write( "\r\n" );
document.write( " b - a = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Square both sides\r\n" );
document.write( "\r\n" );
document.write( " (b-a)^2 = b^2 - a^2,\r\n" );
document.write( "\r\n" );
document.write( " b^2 - 2ab + a^2 = b^2 - a^2\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Cancel b^2 in both sides and continue \r\n" );
document.write( "\r\n" );
document.write( " -2ab + a^2 = - a^2,\r\n" );
document.write( "\r\n" );
document.write( " a^2 + a^2 = 2ab,\r\n" );
document.write( "\r\n" );
document.write( " 2a^2 = 2ab.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Cancel the common factor 2a in both sides\r\n" );
document.write( "\r\n" );
document.write( " a = b.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we proved that under given conditions, it must be a = b.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is what the problem wants to be proved.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER. Under given conditions, the rectangle ABCD is the square and its sides are congruent: a = b.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "It is what the Artificial Intelligence (= @CPhill) missed and could not to prove properly.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |