document.write( "Question 145431: The sum of the lengths of the diagonals of a rhombus of side 61 cm is 142 cm. What is (a) the difference in the lengths of the diagonals, (b) the area of the rhombus? \r
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Algebra.Com's Answer #106364 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
The sum of the lengths of the diagonals of a rhombus of side 61 cm is 142 cm. What is (a) the difference in the lengths of the diagonals, (b) the area of the rhombus? \r
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document.write( "Yes, there's enough information.  First we draw a rhombus with 61cm sides:\r
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document.write( "Now draw in the longer diagonal and label its length x cm.\r
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document.write( "Now take away the bottom half of the rhombus,
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document.write( "and you have an isosceles triangle. Label the
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document.write( "angle \"alpha\":  \r
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document.write( "We use the law of cosines:\r
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document.write( "\"x%5E2=61%5E2%2B61%5E2-2%2861%29%2861%29cos%28alpha%29\" 
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document.write( "\"x+%5E2=7442-7442cos%28alpha%29\"\r
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document.write( "That's one equation we will use.\r
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document.write( "Now let's go back to the original rhombus.\r
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document.write( "Since two angles of a parallelogram
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document.write( "which are next to each other are 
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document.write( "supplementary, we label the angle
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document.write( "\"180\"°\"-alpha\" 
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document.write( "Now we draw the other diagonal and label its length y cm:\r
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document.write( "Take away the right half and we have 
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document.write( "another isosceles triangle.\r
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document.write( "We use the law of cosines again:\r
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document.write( "\"y%5E2=61%5E2%2B61%5E2-2%2861%29%2861%29cos%2890-alpha%29\" 
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document.write( "\"y+%5E2=7442-7442cos%2890-alpha%29\"\r
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document.write( "Now we use the fact that \"cos%2890-alpha%29=-cos%28alpha%29\"\r
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document.write( "\"y+%5E2=7442-7442%28-cos%28alpha%29%29\"\r
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document.write( "or\r
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document.write( "\"y%5E2=7442%2B7442cos%28alpha%29\"\r
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document.write( "Take that equation with the other one:\r
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document.write( "\"x%5E2=7442-7442cos%28alpha%29\"
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document.write( "\"y%5E2=7442%2B7442cos%28alpha%29\"\r
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document.write( "Now we add equals to equals.  Adding
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document.write( "the left sides gives \"x%5E2%2By%5E2\" and
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document.write( "adding the right sides, the cosine terms
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document.write( "cancel out.  So we have:\r
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document.write( "\"x%5E2%2By%5E2=14884\"\r
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document.write( "Now we are told that the sum of the diagonals
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document.write( "is 142 cm., so we have the equation\r
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document.write( "\"x%2By=142\"\r
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document.write( "So we have this system of equations:\r
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document.write( "\"x%5E2%2By%5E2=14884\"
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document.write( "\"x%2By=142\"\r
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document.write( "Can you solve that by solving the second for
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document.write( "one of the letters and substituting into the
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document.write( "first equation?  If not post again asking
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document.write( "how.\r
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document.write( "Solution to that system of equations: \"x=120cm\", \"y=22cm\"\r
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document.write( "Those are the lengths of the diagonals. \r
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document.write( "The difference is just \"120cm-22cm=98cm\".\r
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document.write( "Now since the diagonals of a rhombus are 
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document.write( "perpendicular bisectors of each other, the
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document.write( "shorter diagonal, which is 22 cm is bisected 
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document.write( "by the long diagonal, and so each half is
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document.write( "11 cm. \r
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document.write( "To find the area, take away the bottom half, and
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document.write( "we have this triangle to find the area of:\r
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document.write( "The base of that triangle is \"120cm\", and its
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document.write( "height is \"11cm\"\r
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document.write( "\"A+=+%281%2F2%29bh=%281%2F2%29%28120cm%29%2811cm%29=660cm%5E2\"\r
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document.write( "So we double that to find the total area of the rhombus.\r
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document.write( "Answer = \"1320cm%5E2\"\r
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document.write( "Edwin \n" );
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