Question

The length of the diagonals of a rhombus is in ratio 4 : 3. If its area is 384 cm², find its side.

Answer 1

Let the lengths of the diagonals of a rhombus are 4x, 3x.

∴ Area of the rhombus = ${\displaystyle \frac{1}{2}\times \left(\text{Product of its diagonals}\right)}$

= ${\displaystyle \frac{1}{2}(4x\times 3x)=384}$ (given)

⇒ ${\displaystyle 6x^2=384\Rightarrow x^2=64}$

⇒ x = 8 cm

∴ Diagonals are ${\displaystyle 4\times 8=32}$ cm and ${\displaystyle 3\left(8\right)=24}$ cm.

∴ OC = 16 cm and OD = 12 cm

∴ Side DC = ${\displaystyle \sqrt{{\text{OC}}^2+{\text{OD}}^2}}$

∴ Side DC = ${\displaystyle \sqrt{{16}^2+{12}^2}}$  [By Pythagoras Theorem in ΔDOC]

= ${\displaystyle \sqrt{256+144}=\sqrt{400}=20}$ cm

Hence , side of the rhombus = 20 cm.