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. $\sqrt{3}:1$
• B. $2:\sqrt{5}$
• C. $2:\sqrt{3}$
• D. $\sqrt{5}:2$

Answer 1

The correct option is B $2:\sqrt{5}$

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
$\Rightarrow a^2=\frac{d_1{d}_2}{2}$
Also,
$d_1=2d_2$
$\therefore a^2=\frac{2d_2{d}_2}{2}$
$\Rightarrow a^2=d_2$
$\Rightarrow a=d_2$ and
$\Rightarrow a=\frac{d_1}{2}$
Area of rhombus is half of the product of the diagonals

$\frac{d_1{d}_2}{2}=64$

$\Rightarrow d_1{d}_2=128$
$\Rightarrow \left(2a\right)a=128$
$\Rightarrow a^2=64$
$\Rightarrow a=8cm$
$d_1=2a=8\times 2$
$=16cm$
$d_2=a$
$=8cm$
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 $△$
$\therefore$ Semi-Diagonal $\frac{d_1}{2}=\frac{16}{2}$
$=8cm$
and semi-diagonal $\frac{d_2}{2}=\frac{8}{2}$
$=4cm$
$\therefore 8^2+4^2=b^2$
$\Rightarrow b^2=64+16$
$\Rightarrow b^2=80$
$\Rightarrow b=\sqrt{80}$
$\Rightarrow b=4\sqrt{5}$
$\therefore$ Ratio of perimeter of a square and rhombus $=4a:4b$
$=a:b$
$=8:4\sqrt{5}$
$=2:\sqrt{5}$