7 points are placed in a regular hexagon with side length 20. Let m denote the distance between the two closest points. What is the maximum possible value of m?
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It is 20.
Asits maximum value of m is wanted. So it is 20 as the lenght of side is 20. So, the maximum value is 20.
To find the maximum possible value of m, we need to determine the minimum possible distance between any two points in the hexagon.
In a regular hexagon, there are three types of distances between points:
1. Distance between adjacent points along the sides of the hexagon.
2. Distance between points that are one step apart along the sides of the hexagon.
3. Distance between points that are two steps apart along the sides of the hexagon.
Let's analyze each scenario to find the minimum distances:
1. Distance between adjacent points:
In this case, the distance between adjacent points will be the side length of the hexagon, which is given as 20.
2. Distance between points that are one step apart:
To find this distance, we can draw an equilateral triangle with side length 20, connecting two adjacent points and the center of the hexagon. The distance between points that are one step apart is the side length of this triangle. Using the formula for the length of the altitude of an equilateral triangle, which is (side length * sqrt(3)) / 2, we can calculate the length of the altitude:
Altitude = (20 * √3) / 2 = 10√3
Hence, the distance between points that are one step apart is 10√3.
3. Distance between points that are two steps apart:
To find this distance, we can draw a rectangle that connects two points and has one side aligned with one of the sides of the hexagon. The other side of the rectangle will be equal to two times the distance between adjacent points (2 * 20 = 40). The distance between points that are two steps apart is the length of the rectangle's diagonal.
Using the Pythagorean theorem, we can find the diagonal of the rectangle:
Diagonal = √(20^2 + 40^2) = √(400 + 1600) = √2000 = 20√5
Hence, the distance between points that are two steps apart is 20√5.
Now that we have the three distances, we can determine the minimum possible distance m. Among these distances, the minimum value will be the smallest possible distance between any two points in the hexagon.
So, the minimum possible value of m is the minimum value among the three distances mentioned above, which is 10√3.
Therefore, the maximum possible value of m is 10√3.