document.write( "Question 711640: I have a golden rectangle with a diagonal of 10 cm. What is it width value [x]? No length value is given.\r
\n" ); document.write( "\n" ); document.write( "How do I solve for x?\r
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Algebra.Com's Answer #437564 by KMST(5328)\"\" \"About 
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In a golden rectangle, length \"L\" and width \"W=x\" are related by the ratio
\n" ); document.write( "\"L%2FW=%281%2Bsqrt%285%29%29%2F2\" --> \"L=W%281%2Bsqrt%285%29%29%2F2\" --> \"L=x%281%2Bsqrt%285%29%29%2F2\"
\n" ); document.write( "The diagonal splits the rectangle into 2 congruent right triangles.
\n" ); document.write( "The legs lengths are \"L\" and \"W=x\",
\n" ); document.write( "and the hypotenuse is the diagonal so its length is \"10cm\".
\n" ); document.write( "The Pythagorean theorem says
\n" ); document.write( "\"L%5E2%2BW%5E2=10%5E2\" --> \"%28x%281%2Bsqrt%285%29%29%2F2%29%5E2%2Bx%5E2=100\" --> \"x%5E2%281%2Bsqrt%285%29%29%5E2%2F2%5E2%2Bx%5E2=100\" --> \"x%5E2%281%2B2sqrt%285%29%2B5%29%2F4%2Bx%5E2=100\"
\n" ); document.write( "Multiplying both sides of the equal sign times 4 we get
\n" ); document.write( "\"x%5E2%281%2B2sqrt%285%29%2B5%29%2B4x%5E2=400\" --> \"x%5E2%281%2B2sqrt%285%29%2B5%2B4%29=400\" --> \"x%5E2%282sqrt%285%29%2B10%29=400\" --> \"x%5E2=400%2F%282sqrt%285%29%2B10%29\"
\n" ); document.write( "At this point, we can calculate an approximate value for \"x%5E2\" as a decimal number,
\n" ); document.write( "and then find its square root for a good approximation of \"x\".
\n" ); document.write( "If we want an exact value, all we can get is a more or less complicated expression with square roots:
\n" ); document.write( "\"x%5E2=400%2F%282sqrt%285%29%2B10%29\" --> \"x%5E2=200%2F%28sqrt%285%29%2B5%29\" --> \"x%5E2=200%285-sqrt%285%29%29%2F%28%285-sqrt%285%29%29%285%2Bsqrt%285%29%29%29\"
\n" ); document.write( " --> \"x%5E2=200%285-sqrt%285%29%29%2F%2825-5%29\" --> \"x%5E2=200%285-sqrt%285%29%29%2F20\" --> \"x%5E2=10%285-sqrt%285%29%29\" --> \"x%5E2=50-10sqrt%285%29%29\" --> \"x=sqrt%2850-10sqrt%285%29%29%29\"\r
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