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 = [tex]3 \times Side[/tex]
Let the side be x
So, [tex]81 \sqrt{3} = 3 \times Side[/tex]
[tex]27\sqrt{3} = Side[/tex]
Refer the attached figure .
Since BC = 27√3 cm
BE = half of BC = [tex]\frac{27 \sqrt{3}}{2}[/tex]
All angles of equilateral triangle is 60°
OB is the angle bisector
So, ∠OBC = 30°
In ΔOBE
[tex]tan\theta = \frac{Perpendicular}{Base}[/tex]
[tex]tan30^{\circ} = \frac{OE}{BE}[/tex]
[tex]\frac{1}{\sqrt{3}} = \frac{OE}{\frac{27\sqrt{3}}{2}}[/tex]
[tex]\frac{1}{\sqrt{3}} \times \frac{27\sqrt{3}}{2}=OE[/tex]
[tex]13.5 cm=OE[/tex]
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
[tex]Cos\theta = \frac{Base}{Hypotenuse}[/tex]
[tex]Cos30^{\circ} = \frac{BE}{OB}[/tex]
[tex]\frac{\sqrt{3}}{2} = \frac{\frac{27\sqrt{3}}{2}}{OB}[/tex]
[tex]OB= \frac{\frac{27\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}[/tex]
[tex]OB= 27[/tex]
So, OB = radius =27 cm
