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

There is a typo error, the perimeter of equilateral triangle ABC is 81/√3 centimeters. Answer:Radius = OB= 27 cmApothem = 13.5 cmA diagram is attached for reference.Step-by-step explanation:Given,The perimeter of equilateral triangle ABC is 81/√3 centimeters.Substituting this in the formula of perimeter of equilateral triangle =[tex]3\times\ side[/tex] [tex]3\times\ side[/tex] [tex]=[tex]81\sqrt{3}[/tex][tex]Side = \frac{81\sqrt{3} }{3} =27\sqrt{3} \ cm[/tex]Thus from the diagram , Side [tex]AB=BC=AC= 27\sqrt{3} \ cm[/tex]We know each angle of an equilateral triangle is 60°.From the diagram, OB is an angle bisector.Thus [tex]\angle OBC = 30[/tex]°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.[tex]tan\theta =\frac{perpendicular}{base} \\tan\theta=\frac{OE}{BE} \\tan\theta=\frac{OE}{\frac{27\sqrt{3}}{2}  } \\tan30\times {\frac{27\sqrt{3} }{2} }= OE\\\frac{1}{\sqrt{3} } \times\frac{27\sqrt{3} }{2} =OE\\[/tex]Thus ,apothem  OE= 13.5 cmNow for radius,We consider ΔOBE[tex]cos\theta=\frac{base}{hypotenuse} \\cos30= \frac{BE}{OB} \\Cos30 = \frac{\frac{27\sqrt{3} }{2}}{OB}  \\OB= \frac{\frac{27\sqrt{3} }{2}}{cos30} \\OB= \frac{\frac{27\sqrt{3} }{2}}{\frac{\sqrt{3} }{2} } \\OB =27 \ cm[/tex]Thus for Perimeter of equilateral triangle ABC is 81/√3 centimeters,The radius of equilateral triangle ABC is 27 cmThe apothem of equilateral triangle ABC is 13.5 cm