document.write( "Question 123308This question is from textbook Geometry for Enjoyment and Challenge New Edition
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Algebra.Com's Answer #90493 by solver91311(24713)\"\" \"About 
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document.write( "Discussion
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document.write( "The diagonals of a rhombus are perpendicular bisectors, so half of one diagonal
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document.write( "half of the other diagonal, and one of the sides of the rhombus form a right
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document.write( "triangle with the rhombus side being the hypotenuse.\r
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document.write( "Since we know the perimeter and we know that all four sides of the rhombus are
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document.write( "equal, we can calculate the length of the triangle hypotenuse by dividing the
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document.write( "rhombus perimeter by 4.\r
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document.write( "Since the diagonals mutually bisect each other and the length of the diagonals
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document.write( "are in the ratio 2:1, the measures of the half-diagonals that form the legs of
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document.write( "the triangle must also be in the ratio 2:1.\r
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document.write( "From here, we can apply the Pythagorean theorem to determine the lengths of the
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document.write( "half-diagonals.
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document.write( "Solution
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document.write( "If we let x be a representation of the measure of the short leg of the triangle
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document.write( "then the measure of the long leg must be represented by 2x.\r
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document.write( "The perimeter is given as 20, so the length of one side of the rhombus is:
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document.write( "\"20%2F4=5\"\r
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document.write( "Using Pythagoras,\r
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document.write( "\"sqrt%28%282x%29%5E2%2Bx%5E2%29=5\"
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document.write( "\"sqrt%284x%5E2%2Bx%5E2%29=5\"
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document.write( "\"sqrt%285x%5E2%29=5\"
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document.write( "\"x%2Asqrt%285%29=5\"
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document.write( "\"x=5%2Fsqrt%285%29=sqrt%285%29\"\r
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document.write( "So the measure of the short leg of the triangle is \"sqrt%285%29\" and the 
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document.write( "measure of the short diagonal is \"2%2Asqrt%285%29\".  The measure of the long
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document.write( "leg of the triangle is twice the short leg, or \"2%2Asqrt%285%29\" so the measure
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document.write( "of the long diagonal is \"4%2Asqrt%285%29\".\r
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document.write( "\"2%2Asqrt%285%29+%2B+4%2Asqrt%285%29+=+green%286%2Asqrt%285%29%29\" is then the sum of the lengths of the diagonals.\r
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