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x*sqrt(2)/{5 - sqrt(3)} = {5 + sqrt(3)} => x*sqrt(2) = {5 + sqrt(3)} * {5 - sqrt(3)} = 25 - 3 = 22 => x = 22/sqrt(2) = 22*sqrt(2)/{sqrt(2)*sqrt(2)} = 22*sqrt(2)/2 = 11*sqrt(2)
tan theta = sqrt(2)/2 = 1/sqrt(2).
No
A regular hexagon can be divided into 6 equilateral triangles. If a side equals S, then altitude equals S*sqrt(3)/2, and area of that triangle is (S2)*sqrt(3)/4. So the whole hexagon area is 6*(S2)*sqrt(3)/4 = 3*(S2)*sqrt(3)/2. Since S = perimeter/6, then you have: Area = Perimeter2*sqrt(3)/24.
sqrt(8)/[2*sqrt(3)] = 2*sqrt(2)/[2*sqrt(3)] = sqrt(2)/sqrt(3) = sqrt(2/3)