Q:

The perimeter of equilateral triangle ABC is 81√3 centimeters, find the length of the radius and apothem.The radius of equilateral triangle ABC is _____The apothem of equilateral triangle ABC is _____13.5 cm27 cm81 cm

Accepted Solution

A:
radius = 27 cm apothem = 13.5 cm Since you have an equilateral triangle with a perimeter of 81*sqrt(3), you can just divide the perimeter by 3 to get the length of a side, which is 27*sqrt(3). We can construct a triangle to get the radius as a right triangle with one leg being 27*sqrt(3)/2 in length and the angles being 60/2 = 30 and 180-90-30 = 60 degrees. The tangent of 30 degrees is 1/sqrt(3). So the length of the other leg will be 27*sqrt(3)/2 / sqrt(3) = 27/2. And finally, the hypotenuse will be twice that since it's a 30/60/90 triangle for a length of 27. And conveniently, the hypotenuse is the radius we desire. And also, quite conveniently, the apothem is the short leg of the 30/60/90 triangle which is 27/2 = 13.5 cm.