CAT Exam Model Paper 5 with solutions for free online practice

Let the hexagon be ABCDEF. The rectangle formed by joining the mid-points P, Q, R and S is shown below:

Let FM ⊥ SR and AN ⊥ PS
AS = SF = $\frac{\mathrm{AF}}{2}$ = a units
In right angled ∆SMF:
SM = SF cos60° = $\frac{\mathit{a}}{2}$
Hence, SR = 2a + 2SM = 3a
Similarly in right angled ∆ANS :
NS = AS sin 60° = $\frac{\sqrt{3}\mathit{a}}{2}$
Hence, PS = 2NS =$\sqrt{3}$a
Area of rectangle PQRS = PS × SR = $3\sqrt{3}{\mathit{a}}^2$
Area of hexagon PQRS = $6\mathrm{}\frac{\sqrt{3}}{4}\left(4{\mathit{a}}^2\right)=6\sqrt{3}{\mathit{a}}^2$
∴Ratio $=3\sqrt{3}{\mathit{a}}^2:6\sqrt{3}{\mathit{a}}^2=1:2$