CAT 2015 Quantitative Ability

PQRS is a rhombus, so the centre of the inscribed circle will be the center of the rhombus PQRS
SQ = 1/2(AB + CD)= 1/2(2+14) = 8 units
OR = 1/2 PR
DU = 1/2(CD - MU) = 1/2(14 - AB) = 6 units
$\mathit{P}\mathit{R}=\mathit{A}\mathit{U}=\sqrt{{\mathit{A}\mathit{D}}^2-{\mathit{D}\mathit{U}}^2}=\sqrt{{10}^2-6^2}=8\mathit{u}\mathit{n}\mathit{i}\mathit{t}\mathit{s}$
PR= SQ = 8
⇒PQRS is a square.
⇒ OT = $2\sqrt{2}$ units
Area of trapezium = $\frac{1}{2}\left(\mathit{A}\mathit{B}+\mathit{C}\mathit{D}\right)\times \mathit{P}\mathit{R}=64$ sq.unit
Required ratio = $\frac{\mathit{\pi }\mathit{O}{\mathit{T}}^2}{64}=\frac{\mathit{\pi }\times 8}{64}=\frac{\mathit{\pi }}{8}$