A regular hexagon is composed of 12 congruent 30-60-90 triangles. If the length of the hypotenuse of one of those triangles is 18 square root 3, find the perimeter of the hexagon.
sin30° = 1/2
half a side is thus 9√3
so, 12 sides will be ???
To find the perimeter of the hexagon, we need to determine the length of each side of the hexagon.
Since the given triangle is a 30-60-90 triangle, we can use the ratios of the sides in this type of triangle: the sides are in the ratio 1:√3:2.
In this case, the hypotenuse is 18√3, which corresponds to the longest side in the 30-60-90 triangle. According to the ratio, this side is twice the length of the shorter side.
So, we can find the length of the shorter side by dividing the hypotenuse by 2:
Shorter side length = (18√3) / 2 = 9√3
Since each triangle makes up one-sixth of the hexagon, we need to multiply the side length of the triangle by 6 to find the perimeter of the hexagon:
Perimeter of hexagon = 6 * Shorter side length = 6 * 9√3
Simplifying, we have:
Perimeter of hexagon = 54√3
Therefore, the perimeter of the hexagon is 54√3.