document.write( "Question 1113279: Circular logic? (H). Take a Golden Rectangle and draw the largest circle
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Algebra.Com's Answer #728383 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The answer to problem as stated is no.  However there is a\r\n" );
document.write( "similar problem to which the answer is yes. First the problem\r\n" );
document.write( "as stated is no:\r\n" );
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document.write( "This figure below is drawn to scale. The longer side of the\r\n" );
document.write( "large rectangle is \"phi=%28%281%2Bsqrt%285%29%29%2F2%29\" times the shorter\r\n" );
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document.write( "  \r\n" );
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document.write( "All golden rectangles are similar.  So the answer is \r\n" );
document.write( "no, because neither of those two rectangles is similar\r\n" );
document.write( "to the large rectangle.\r\n" );
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document.write( "Now here is a similar problem where the answer is yes.\r\n" );
document.write( "We draw the line different: \r\n" );
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\n" ); document.write( "Take a Golden Rectangle and draw the largest circle
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document.write( "The big rectangle below is Golden.  The short side is \"a\"\r\n" );
document.write( "and the long side is \"a%2Aphi\".  The circle is inscribed\r\n" );
document.write( "in a square. The bottom side of the square is \"a\" units\r\n" );
document.write( "So the short side of the small rectangle on the right is \r\n" );
document.write( "\"a%2Aphi-a\" \r\n" );
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document.write( "                          \"a%2Aphi\"\r\n" );
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document.write( "Now we will show that the small rectangle on the right is golden.\r\n" );
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document.write( "The ratio of the sides is\r\n" );
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document.write( "\"a%2F%28a%2Aphi-a%29\"\r\n" );
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document.write( "Divide numerator and denominator by a\r\n" );
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document.write( "\"1%2F%28phi-1%29\"\r\n" );
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document.write( "Substitute \"phi=%28%281%2Bsqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "\"1%5E%22%22%2F%28%281%2Bsqrt%285%29%29%2F2-1%29\"\r\n" );
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document.write( "Multiply top and bottom by 2\r\n" );
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document.write( "\"2%2F%28%281%2Bsqrt%285%29%29-2%29\"\r\n" );
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document.write( "\"2%2F%281%2Bsqrt%285%29-2%29\"\r\n" );
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document.write( "\"2%2F%28-1%2Bsqrt%285%29%29\"\r\n" );
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document.write( "Rationalize the denominator by multiplying by \"%28-1-sqrt%285%29%29%2F%28-1-sqrt%285%29%29\"\r\n" );
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document.write( "\"2%2F%28-1%2Bsqrt%285%29%29\"\"%22%22%2A%22%22\"\"%28-1-sqrt%285%29%29%2F%28-1-sqrt%285%29%29\"\r\n" );
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document.write( "\"%28-2-2sqrt%285%29%29%2F%281-5%29%29\"\r\n" );
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document.write( "\"%28-2-2sqrt%285%29%29%2F%28-4%29%29\"\r\n" );
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document.write( "\"%28-2%281%2Bsqrt%285%29%29%29%2F%28-4%29%29\"\r\n" );
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document.write( "Divide top and bottom by -2:\r\n" );
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document.write( "\"%281%2Bsqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "Which is \"phi\", the golden ratio.\r\n" );
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document.write( "So for this problem, the answer is \r\n" );
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document.write( "Yes, the small rectangle on the right is golden.\r\n" );
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document.write( "[I have a hunch this was the problem that was intended.\r\n" );
document.write( "Point this out to your teacher.]\r\n" );
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document.write( "Edwin

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