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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
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B
2:5
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C
2:3
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D
5:2
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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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