the perimeter of the dodecagon is 54 cm. by joining every other vertex of the dodecagon a hexagon is formed. if each side of the hexagon is x cm long, what is the value of the x to the nearest hundredth

x = 8.69

Step-by-step explanation:

we know that the perimeter of the dodecagon is 54, so each edge will be 54/12

54/12 = 4.5 cm

if we draw the lines to remove 6 vertices and form a hexagon, 6 triangles with 2 sides of 4.5 cm are formed.

we know that the angle of each vertex is 150 ° because it is a dodecagon

if we apply the law of cosines we can take the other side of the triangle, since we only need 2 side and the opposite angle to the side we want to know

a would be our x

b = 4.5

c = 4.5

A = 150°

a^2 = b^2 + c^2 - 2bc * cos (A)

x^2 = 4.5^2 + 4.5^2 - 2 * 4.5 * 4.5 * cos (150)

x^2 = 20.25 + 20.25 - 40.50 * - 0.866

x^2 = 40.50 + 35.07

x = √ 75.57

x = 8.69