ABCDEF is a regular hexagon with side length 43√4. 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?
Sorry,the side length is 4*(3)^1/4
54
Explain, please?
To find the expected area of hexagon GHIJKL, we need to calculate the average area over all possible positions of points G, H, I, J, K, and L.
Let's start by finding the area of hexagon ABCDEF. A regular hexagon can be divided into six equilateral triangles. Each triangle has side length equal to the side length of the hexagon.
Since the side length of hexagon ABCDEF is given as 43√4, the side length of each equilateral triangle is also 43√4.
The area of an equilateral triangle can be calculated using the formula:
A = (√3 / 4) * s^2
Where A is the area and s is the length of a side.
Plugging in the values, the area of each equilateral triangle is:
A_triangle = (√3 / 4) * (43√4)^2 = (√3 / 4) * (43^2 * 4) = (√3 / 4) * (43^2) * 4 = 43^2 * (√3 / 4)
Since there are six equilateral triangles making up hexagon ABCDEF, the total area of the hexagon is:
A_hexagon = 6 * A_triangle = 6 * (43^2 * (√3 / 4))
Now, let's find the expected area of hexagon GHIJKL by considering the possible positions of points G, H, I, J, K, and L.
Since each of these points is chosen uniformly at random from one side of the hexagon, the average position of each point will be at the midpoint of its corresponding side.
Therefore, hexagon GHIJKL will be an irregular hexagon with side lengths equal to half the side lengths of hexagon ABCDEF.
The side length of hexagon GHIJKL becomes (43√4) / 2 = 43√4 / 2 = 21.5√4
To calculate the area of hexagon GHIJKL, we can use the formula for irregular hexagons:
A = (3√3 / 2) * s^2
Where A is the area and s is the length of a side.
Plugging in the values, the area of hexagon GHIJKL is:
A_GHIJKL = (3√3 / 2) * (21.5√4)^2 = (3√3 / 2) * (21.5^2 * 4) = (3√3 / 2) * (21.5^2) * 4 = 21.5^2 * (√3 / 2)
Finally, to find the expected area of hexagon GHIJKL, we need to average the areas of all possible positions of points G, H, I, J, K, and L. Since each point is chosen uniformly at random, the expected area is the average area.
Therefore, the expected area of hexagon GHIJKL is:
Expected_Area_GHIJKL = A_GHIJKL = 21.5^2 * (√3 / 2)
Calculating this value will provide the answer to the question.