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.
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 cm
Now 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 cm
The apothem of equilateral triangle ABC is 13.5 cm
