question_answer

ABCDEF is a regular hexagon of side 2 feet. The area, in square feet, of the rectangle BCEF is

A)  8                                

B)  \[4+4\sqrt{3}\]

C)  4                                

D)  \[4\sqrt{3}\]

Correct Answer: D

Solution :

Each interior angle \[=\left( \frac{2n-4}{n} \right)90{}^\circ \] \[=\frac{2\times 6-4}{6}\times 90{}^\circ =120{}^\circ \] \[\therefore \]\[\angle FAM=60{}^\circ \] \[\angle AFM=30{}^\circ \] From \[\Delta \]AMF, FM=AF cos \[30{}^\circ \] \[=2\times \frac{\sqrt{3}}{2}=\sqrt{3}\]feet \[\therefore \]BF = \[2\sqrt{3}\]feet, BC = 2 feet \[\therefore \] Area of rectangle \[=2\times 2\sqrt{3}\]\[=4\sqrt{3}\] sq. feet