SSC CHSL 10 Jan 2017 Shift 3 Paper

AC and BD are diagonals of the rhombus ABCD with side = 10 cm
One diagonal of a rhombus is half the other, let AC = 4x cm
⇒ BD = 2x cm
Diagonal of a rhombus perpendicularly bisect each other.
⇒ OD = x cm and OC = 2x cm
In ∆OCD, ⇒ ${\left(\mathit{O}\mathit{C}\right)}^2+{\left(\mathit{O}\mathit{D}\right)}^2={\left(\mathit{C}\mathit{D}\right)}^2$
⇒ ${\left(2\mathit{x}\right)}^2+{\left(\mathit{x}\right)}^2={\left(10\right)}^2$
⇒ $4{\mathit{x}}^2+{\mathit{x}}^2=100$
⇒ ${\mathit{x}}^2=\frac{100}{5}=20$
∴ Area of rhombus = $\frac{1}{2}\times$ Product of diagonals
= $\frac{1}{2}\times 4\mathit{x}\times 2\mathit{x}$
= $4{\mathit{x}}^2=4\times 20=80\mathit{c}{\mathit{m}}^2$