a regular hexagon is composed of 12 congruent 30,60,90 degree triangles. if the length of the hypotenuse of one of those triangles is 18 square root 3 find the perimeter of the hexagon
To find the perimeter of the hexagon, we need to know the length of one side of the hexagon.
Since each of the 30-60-90 degree triangles in the hexagon are congruent, and the hypotenuse of one of the triangles is given as 18√3, we can find the length of one side of the triangle by using the ratios of the sides in a 30-60-90 triangle.
The ratio of the sides in a 30-60-90 triangle is 1 : √3 : 2. Therefore, the length of the hypotenuse (18√3) corresponds to the value of 2 in the ratio.
Let's set up the proportion:
18√3 / 2 = x / 1
Cross-multiplying, we get:
36√3 = x
Simplifying, we find:
x = 36√3
So, the length of one side of the hexagon is 36√3.
Now, to find the perimeter of the hexagon, we need to multiply the length of one side by 6 (since a hexagon has 6 sides):
Perimeter = 6 * 36√3
Perimeter = 216√3
Therefore, the perimeter of the hexagon is 216√3 units.
To find the perimeter of the hexagon, we need to determine the length of one side of the hexagon and multiply it by 6.
Each 30-60-90 degree triangle has a hypotenuse of 18√3. In a 30-60-90 degree triangle, the hypotenuse is twice the shorter side (which is the side opposite the 30-degree angle).
So, the length of the shorter side of the triangle is (18√3)/2 = 9√3.
In the regular hexagon, there are six shorter sides that form the perimeter. Therefore, the perimeter of the hexagon is:
Perimeter = 6 * (9√3) = 54√3.
Thus, the perimeter of the hexagon is 54√3 units.
let x be the base of one of those triangles.
cos 60° = x/(18√3)
x = 18√3 cos 60°
= 18√3(√3/2) = 9
but x would be 1/2 of one of the sides of the hexagon
so each side is 18
and the perimeter = 6(18) or 108