show that the length of each side of regular hexagon is equal to the length of the radius of the hexagon.
A regular hexagon can be divided into 12 congruent equilateral triangles by drawing lines from the center of the hexagon to each vertex and the midpoint of each side.
Let's consider one of these equilateral triangles. Let the side length of the hexagon (and triangle) be x, and let the length of the radius of the hexagon (which is also the height of the equilateral triangle) be r.
Since the equilateral triangle is symmetric, we can bisect it to form a 30-60-90 right triangle with hypotenuse r, one leg being half the side length of the hexagon (x/2), and the other leg being the height of the bisected equilateral triangle (let's call this height h).
In a 30-60-90 right triangle, the ratios of the sides are 1 : sqrt(3) : 2. This means that the ratio of the smallest side (x/2) to the largest side (r) is 1/2, and the ratio of the height (h) to the hypotenuse (r) is sqrt(3)/2. Using these ratios, we can write:
x/2 = (1/2)r => x = r
So, the length of each side of the regular hexagon (x) is equal to the length of the radius of the hexagon (r).