  Question

Areas of square and rhombus are equal. A diagonal of a rhombus is twice of its other diagonal. If the area of rhombus is $$64$$ sq. cm find the ratio of perimeter of a square and rhombus.

A
3:1  B
2:5  C
2:3  D
5:2  Solution

The correct option is B $$2 : \sqrt5$$Let $$a$$ be the side of a square and$$b$$ be the side of a rombus,and $$d_1, d_2$$ be the two diagonals of the rombus.Given,Area of square $$=$$ Area of rombus$$\Rightarrow a^2=\dfrac{d_1 d_2}{2}$$Also,$$d_1=2d_2$$$$\therefore a^2=\dfrac{2d_2 d_2}{2}$$$$\Rightarrow a^2=d_2$$$$\Rightarrow a=d_2$$ and$$\Rightarrow a=\cfrac{d_1}{2}$$Area of rhombus is half of the product of the diagonals$$\cfrac{d_1 d_2}{2}=64$$$$\Rightarrow d_1 d_2=128$$$$\Rightarrow (2a)a=128$$$$\Rightarrow a^2=64$$$$\Rightarrow a=8\ cm$$$$d_1=2a=8\times 2$$      $$=16\ cm$$$$d_2=a$$     $$=8\ cm$$Diagonals of a rhombus bisect each other at right angles. So, half of both the diagonals and the side of the rhombus make a right angled $$\triangle$$$$\therefore$$ Semi-Diagonal $$\dfrac{d_1}{2}=\dfrac{16}{2}$$                                    $$=8\ cm$$and semi-diagonal $$\dfrac{d_2}{2}=\dfrac82$$                                       $$=4\ cm$$$$\therefore 8^2+4^2=b^2$$$$\Rightarrow b^2=64+16$$$$\Rightarrow b^2=80$$$$\Rightarrow b=\sqrt{80}$$$$\Rightarrow b=4\sqrt5$$$$\therefore$$ Ratio of perimeter of a square and rhombus $$=4a:4b$$                                                                              $$=a:b$$                                                                              $$=8: 4\sqrt5$$                                                                              $$=2:\sqrt 5$$Maths

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