Question 1043916: A circle is inscribed in an equilateral triangle. If the circumference of it is 1cm find the exact perimeter of this triangle.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The center of a circle inscribed in a triangle is called the incenter of the triangle. The incenter is located at the intersection of the bisectors of the three angles of the triangle.
Refer to the diagram
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BD bisects angle ABC, AE bisects angle BAC. Since ABC is equilateral, BD is the perpendicular bisector of AC, and AE is the perpendicular bisector of BC.
All angles of an equilateral triangle measure 60 degrees, so angle OAD must measure 30 degrees. Since BD perpendicular to AC, angle ODA measures 90 degrees, then angle AOD measures 60 degrees perforce.
Since the inscribed circle is tangent at D, D is on the circle, hence OD is a radius of the circle. Given the circumference is 1 cm, the radius of the circle must be  cm.
Since the sides of a 30:60:90 right triangle are in proportion  , segment AD must measure  cm. Since segment AD is 1/6 of the perimeter of the equilateral triangle, the entire perimeter must be:
 cm.
John
My calculator said it, I believe it, that settles it
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