Question

The perimeter of equilateral triangle ABC is 81√3 centimeters, find the length of the radius and apothem.

Answer 1

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.

Answer 2

Answer:

OB = radius =27 cm

Apothem of equilateral triangle ABC is 13.5 cm.

Step-by-step explanation:

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

Perimeter of equilateral triangle = $3 \times Side$

Let the side be x

So,  $81 \sqrt{3} = 3 \times Side$

$27\sqrt{3} = Side$

Refer the attached figure .

Since BC = 27√3 cm

BE = half of BC = $\frac{27 \sqrt{3}}{2}$

All angles of equilateral triangle is 60°

OB is the angle bisector

So, ∠OBC = 30°

In ΔOBE

$tan\theta = \frac{Perpendicular}{Base}$

$tan30^{\circ} = \frac{OE}{BE}$

$\frac{1}{\sqrt{3}} = \frac{OE}{\frac{27\sqrt{3}}{2}}$

$\frac{1}{\sqrt{3}} \times \frac{27\sqrt{3}}{2}=OE$

$13.5 cm=OE$

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

$Cos\theta = \frac{Base}{Hypotenuse}$

$Cos30^{\circ} = \frac{BE}{OB}$

$\frac{\sqrt{3}}{2} = \frac{\frac{27\sqrt{3}}{2}}{OB}$

$OB= \frac{\frac{27\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}$

$OB= 27$

So, OB = radius =27 cm