document.write( "Question 333740: The Diagonals of Rhombus measure 16cm and 30cm.Find its perimeter \n" ); document.write( "
Algebra.Com's Answer #239188 by Edwin McCravy(20059) $\"\"$ $\"About$
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document.write( "Here is the rhombus drawn twice, once with each diagonal.  The four\r\n" );
document.write( "sides of a rhombus have equal measures. Let each side have length x.\r\n" );
document.write( "Then the perimeter of the rhombus will be 4x.\r\n" );
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document.write( "Looking at the lower triangular half of the left drawing we use the \r\n" );
document.write( "law of cosines on triangle ABC:\r\n" );
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document.write( "Divide every term by 2\r\n" );
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document.write( "Now we look at the left triangular half of the right drawing, and\r\n" );
document.write( "we use the law of cosines again, this time on triangle ABD:\r\n" );
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document.write( "Divide every term by 2\r\n" );
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document.write( "Here are the two equations we have found above:\r\n" );
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document.write( "A rhombus is a parallelogram, and the adjacent angles in a\r\n" );
document.write( "parallelogram are supplementary.  Therefore the sum of the\r\n" );
document.write( "measures of angles A and B is 180°.  Therefore\r\n" );
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document.write( "B = 180° - A\r\n" );
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document.write( "Therefore we use the identity:  ,\r\n" );
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document.write( "and get: \r\n" );
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document.write( "cos(B) = cos(180°-A) = -cos(A) \r\n" );
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document.write( "We substitute -cos(A) for cos(B) in the first equation and simplify:\r\n" );
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document.write( "Now we put that together with the other equation and we have this system:\r\n" );
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document.write( "When we add those two equations term-by term, the terms on the right cancel\r\n" );
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document.write( "and we get:\r\n" );
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document.write( "Divide both sides by 2\r\n" );
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document.write( "Taking positive square roots of both sides:\r\n" );
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document.write( "So the perimeter is 4x or 4(17) or 68.\r\n" );
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document.write( "Edwin
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