document.write( "Question 262990: A square with side length s is inscribed as in an equilateral triangle with side length t. What is the ratio t : s? Express your answer as a decimal to the nearest thousandth. \n" ); document.write( "
Algebra.Com's Answer #193996 by Edwin McCravy(20086)\"\" \"About 
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document.write( "Let the equilateral triangle be ABC, and the\r\n" );
document.write( "inscribed square be DEFG:\r\n" );
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document.write( "We draw in the green median CH from vertex C to the bottom\r\n" );
document.write( "side AB, which is also the perpendicular bisector of AB,\r\n" );
document.write( "and also the bisector of the angle C.\r\n" );
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document.write( "We find the length of CH by the Pythagorean theorem\r\n" );
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document.write( "\"AC+=+t\", \"AH+=+t%2F2\"\r\n" );
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document.write( "\"AC%5E2=AH%5E2%2BCH%5E2\"\r\n" );
document.write( "\"t%5E2=%28t%2F2%29%5E2%2BCH%5E2\"\r\n" );
document.write( "\"t%5E2=t%5E2%2F4%2BCH%5E2\"\r\n" );
document.write( "\"4t%5E2=t%5E2%2B4%2ACH%5E2\"\r\n" );
document.write( "\"3t%5E2=4%2ACH%5E2\"\r\n" );
document.write( "\"%283t%5E2%29%2F4=CH%5E2\"\r\n" );
document.write( "\"sqrt%28%283t%5E2%29%2F4%29=sqrt%28CH%5E2%29\"\r\n" );
document.write( "\"%28t%2Asqrt%283%29%29%2F2=CH\"\r\n" );
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document.write( "Triangle ADG is similar to triangle AHC,\r\n" );
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document.write( "so \"%28GD%29%2F%28AD%29=%28CH%29%2F%28AH%29\"\r\n" );
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document.write( "\"AH+=t%2F2\", \"DH=s%2F2\", so\r\n" );
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document.write( "\"AD=AH-DH=t%2F2-s%2F2=%28t-s%29%2F2\" \r\n" );
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document.write( "\"GD=s\", \"CH=%28t%2Asqrt%283%29%29%2F2\"\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "\"%28GD%29%2F%28AD%29=%28CH%29%2F%28AH%29\"\r\n" );
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document.write( "\"s%2F%28%28t-s%29%2F2%29=%28%28t%2Asqrt%283%29%29%2F2%29%2F%28t%2F2%29\"\r\n" );
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document.write( "Multiply tops and bottoms of the fractions on both\r\n" );
document.write( "sides by 2.\r\n" );
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document.write( "\"%282s%29%2F%28t-s%29=%28t%2Asqrt%283%29%29%2Ft\"   \r\n" );
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document.write( "Cancel t's on the right sides:\r\n" );
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document.write( "\"%282s%29%2F%28t-s%29=sqrt%283%29\"\r\n" );
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document.write( "Multiply both sides by \"%28t-s%29\"\r\n" );
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document.write( "\"2s=sqrt%283%29%28t-s%29\"\r\n" );
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document.write( "\"2s=sqrt%283%29t-sqrt%283%29s\"\r\n" );
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document.write( "\"2s%2Bsqrt%283%29s=sqrt%283%29t\"\r\n" );
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document.write( "factor out s on the left:\r\n" );
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document.write( "\"s%282%2Bsqrt%283%29%29=sqrt%283%29t\"\r\n" );
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document.write( "Divide both sides by \"s%2Asqrt%283%29\"\r\n" );
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document.write( "\"%282%2Bsqrt%283%29%29%2Fsqrt%283%29=t%2Fs\"\r\n" );
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document.write( "Punching the left side out on a calculator\r\n" );
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document.write( "\"t%2Fs=2.154700538\"\r\n" );
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document.write( "To the nearest thousandth, \r\n" );
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document.write( "\"t%2Fs=2.155\"\r\n" );
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document.write( "Edwin
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