Question

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 cm
27 cm
81 cm

Answer 1

OB = radius =27 cm

Apothem of equilateral triangle ABC is 13.5 cm.

Explanation:

The perimeter of equilateral triangle ABC is 81√3 centimeters.

Perimeter of equilateral triangle =
`3 * Side`

Let the side be x

So,
`81 √(3) = 3 * Side`

`27√(3) = Side`

Refer the attached figure .

Since BC = 27√3 cm

BE = half of BC =
`(27 √(3))/(2)`

All angles of equilateral triangle is 60°

OB is the angle bisector

So, ∠OBC = 30°

In ΔOBE

`tan\theta = (Perpendicular)/(Base)`

`tan30^(\circ) = (OE)/(BE)`

`(1)/(√(3)) = (OE)/((27√(3))/(2))`

`(1)/(√(3)) * (27√(3))/(2)=OE`

`13.5 cm=OE`

The apothem is equivalent to the line segment from the midpoint of a side to any of the triangle's centers

So, OE is apothem

Thus The apothem of equilateral triangle ABC is 13.5 cm.

Now In ΔOBE

`Cos\theta = (Base)/(Hypotenuse)`

`Cos30^(\circ) = (BE)/(OB)`

`(√(3))/(2) = ((27√(3))/(2))/(OB)`

`OB= ((27√(3))/(2))/((√(3))/(2))`

`OB= 27`

So, OB = radius =27 cm

Answer 2

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.