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The side lengths of a rhombus and a square are same. But area of square is twice the area of rhombus. What will be the ratio of lengths of diagonals of rhombus?
The side lengths of a rhombus and a square are same. But area of square is twice the area of rhombus. What will be the ratio of lengths of diagonals of rhombus?
1). (2 + √3) : 2
2). (2 + √3) : 1
3). √3 : 2
4). √3 : 1
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Let the length of diagonals of rhombus be M and N.
⇒ Side length of square = Sid length of rhombus = √ ((M/2)2 + (N/2)2) = (1/2) √ ((M)2 + (N)2)
Area of rhombus = MN/2
Area of square = square of (1/2) √ ((M)2 + (N)2) = ((M)2 + (N)2)/4
As per given condition, ((M)2 + (N)2)/4 = 2MN/2 = MN
((M)2 + (N)2) = 4MN
Divide both side by N2, and take M/N = T, we get
T2 – 4T + 1 = 0
Solving, we get T = M/N = 2 + √3, 2 - √3
∴ Ratio of lengths is 2 + √3, among given options.