In the figure above, a regular hexagon EFGHIJ is inscribed in the rect

Some redundant info in the question...

Check the image below:



Each interior angle of a regular polygon is given by: \(\frac{180(n−2)}{n}\), where n is the number of sides. Thus, each interior angel in the regular hexagon = 120°. This will make triangles at the corner 30°-60°-90°. The sides are always in the ratio 1:√3:2. Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°).

So, we have that one side of the rectangle is 1 + 2 + 1 = 4 and another is \(\sqrt{3}+\sqrt{3}=2\sqrt{3}\)

Area = \(4*2\sqrt{3}=8\sqrt{3}\);

Perimeter = \(2*(4+2\sqrt{3})=8+4\sqrt{3}\).

Ratio = \(\frac{8\sqrt{3}}{8+4\sqrt{3}}=\frac{2\sqrt{3}}{2+\sqrt{3}}\).

Rationalize by multiplying numerator and denominator by \(2-\sqrt{3}\):

\(\frac{2\sqrt{3}(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}=\frac{2\sqrt{3}(2-\sqrt{3})}{4-1}=4\sqrt{3}-6\).

Answer: B.