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):
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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
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).
