ABCDEF is a regular hexagon with side length 4 (Under 4 root)√3. Points G,H,I,J,K,L are chosen uniformly at random, one from each side of the hexagon. What is the expected area of hexagon GHIJKL?

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To find the expected area of hexagon GHIJKL, we first need to calculate the area of each possible hexagon that can be formed by choosing points on each side of the given hexagon ABCDEF.

Let's start by calculating the area of hexagon GHIJKL. As ABCDEF is a regular hexagon with side length 4√3, each side of GHIJKL will also be 4√3.

Now, let's divide GHIJKL into three parts: triangles GHI, JIK, and LKG.

Each of these triangles is an equilateral triangle with side length 4√3, so we can calculate the area of each triangle using the formula:

Area of an equilateral triangle = (√3 / 4) * (side length)^2

Therefore, the area of each triangle is:

Area of GHI = (√3 / 4) * (4√3)^2 = (√3 / 4) * (4*4*3) = (√3 / 4) * (48) = 12√3

Area of JIK = 12√3

Area of LKG = 12√3

Next, let's calculate the area of hexagon GHIJKL by summing up the areas of the three triangles:

Area of GHIJKL = Area of GHI + Area of JIK + Area of LKG
= 12√3 + 12√3 + 12√3
= 36√3

Since each of the triangles has the same area, the total area of hexagon GHIJKL is 36√3.

Finally, to find the expected area, we need to consider that each point (G, H, I, J, K, and L) can be chosen uniformly at random from each side of the hexagon. Therefore, the expected area is the average of all possible areas of GHIJKL.

Since there are 6 sides of the hexagon and each side has an area of 36√3, the expected area is given by:

Expected area = (Area of GHIJKL) / (Number of sides of the hexagon)
= 36√3 / 6
= 6√3

Hence, the expected area of hexagon GHIJKL is 6√3.