SOLUTION: A rhombus has half the area of the square with the same side-length. What is the ratio of the long to short diagonal?

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Question 1127404: A rhombus has half the area of the square with the same side-length. What is the ratio of the long to short diagonal?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

To find the area of a rhombus use:

if the sides of a rhombus is , the area of a square is
if the area of a rhombus equal to the area of a square, we have




Using Pythagoras: triangle formed bu side and half of diagonals


.......(2)
Hence the direct relation between diagonals are not possible.
Properties of Rhombus

Answer by ikleyn(52855)   (Show Source): You can put this solution on YOUR website!
.
Let x be the common length of the square and the rhombus.


Use the formula for the rhombus area 

    Rhombus area =  = ,

where  is the angle between (any) two adjacent sides of the rhombus.



Comparing with the formula for the square area, you can conclude that  = ;

hence,  the angle     is either  30° or 150°, which geometrically represent the same rhombus.



Then the longer diagonal of the rhombus is  ( the law of cosine with cos(150°) =  )

     =  =  = .


The shorter diagonal is ( the law of cosine with cos(30°) =  )

     =  =  = .



Thus the ratio of the longer diagonal length to the shorter diagonal length is  .


You can rationalize this fraction further

     = . =  =  =  = 3.732 (approximately).

Solved.


Nice solution to a nice problem.



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