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?
To solve this problem, let's consider the points in the hexagon. A regular hexagon has six vertices, so there are six points in total.
To find the maximum possible value of m, we need to find the minimum distance between any two points in the hexagon.
Since the hexagon is regular, we can divide it into six equilateral triangles. Each side of the hexagon is the same length, in this case, 20 units. Therefore, each side of the equilateral triangles formed is also 20 units.
Now, let's investigate the distances between the points in the hexagon. Each point is connected to two adjacent points by a side of the hexagon. These distances are all 20 units since they form the sides of the hexagon.
To find the minimum distance between any two points, we need to consider the diagonals of the hexagon. Each point is separated from the three non-adjacent points by a diagonal. The diagonals of a regular hexagon create another smaller equilateral triangle inside the hexagon.
The sides of this smaller equilateral triangle are also 20 units long, but now we are measuring the distance between non-adjacent points. The distances between non-adjacent points in this smaller equilateral triangle are equal to the side length of the larger equilateral triangle (20 units) minus the side length of the smaller equilateral triangle (√3 * 20 / 2).
So the minimum distance between any two points in the hexagon is equal to (20 - √3 * 20 / 2) units.
Finally, to find the maximum possible value of m, we need to find the greatest minimum distance between any two points.
Calculating (20 - √3 * 20 / 2) gives us approximately 8.6603 units.
Therefore, the maximum possible value of m is approximately 8.6603 units.