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

Answer 1

OB = 27 cm

Explanation:

Answer 2

There is a typo error, the perimeter of equilateral triangle ABC is 81/√3 centimeters.

Answer:

Radius = OB= 27 cm

Apothem = 13.5 cm

A diagram is attached for reference.

Explanation:

Given,

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

Substituting this in the formula of perimeter of equilateral triangle =
`3*\ side`

`3*\ side`
`=[tex]81√(3)`

`Side = (81√(3) )/(3) =27√(3) \ cm`

Thus from the diagram , Side
`AB=BC=AC= 27√(3) \ cm`

We know each angle of an equilateral triangle is 60°.

From the diagram, OB is an angle bisector.

Thus
`\angle OBC = 30`°

Apothem is the line segment from the mid point of any side to the center the equilateral triangle.

Therefore considering ΔOBE, and applying tan function.

`tan\theta =(perpendicular)/(base) \\tan\theta=(OE)/(BE) \\tan\theta=(OE)/((27√(3))/(2)  ) \\tan30* {(27√(3) )/(2) }= OE\\(1)/(√(3) ) *(27√(3) )/(2) =OE\\`

Thus ,apothem OE= 13.5 cm

Now for radius,

We consider ΔOBE

`cos\theta=(base)/(hypotenuse) \\cos30= (BE)/(OB) \\Cos30 = ((27√(3) )/(2))/(OB)  \\OB= ((27√(3) )/(2))/(cos30) \\OB= ((27√(3) )/(2))/((√(3) )/(2) ) \\OB =27 \ cm`

Thus for

Perimeter of equilateral triangle ABC is 81/√3 centimeters,

The radius of equilateral triangle ABC is 27 cm

The apothem of equilateral triangle ABC is 13.5 cm