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?
Don't answer, this is a Brilliantorg question for the week,
20
To find the maximum possible value of m, we can consider the arrangement of the 7 points in the hexagon.
Since a regular hexagon has 6 equal sides, we can divide it into 6 equilateral triangles. Each equilateral triangle formed by the hexagon has angles of 60 degrees.
Now, let's consider the arrangement of the 7 points:
1. We place one point in the center of the hexagon.
2. The remaining 6 points are placed at the vertices of the hexagon.
Since the hexagon is regular, all the distances from the center to the vertices are equal and have a length of 20.
Next, we need to find the distance m between the two closest points. To do this, we can consider the two points that are closest to each other. These points will be adjacent vertices of the hexagon.
Let's label the points as A, B, C, D, E, F (with the center point being O). Points A and B are adjacent vertices. To find the distance AB, we can use the fact that AB is equal to twice the height of an equilateral triangle with side length 20 as it is the distance of two vertices of the hexagon that are directly across from each other.
The height of an equilateral triangle with side length 20 can be found using the formula:
h = (√3 / 2) * s, where s is the side length of the equilateral triangle.
Substituting the value of s = 20:
h = (√3 / 2) * 20 = 10√3.
Therefore, the distance AB (or m) is equal to twice the height:
m = 2 * 10√3 = 20√3.
Hence, the maximum possible value of m is 20√3.