An equilateral triangle is circumscribed about a circle of radius 10sqrt3. What is the perimeter of the triangle?

1 Answer
Sep 28, 2015

The perimeter of the triangle is 180 units.


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In the above image equilateral triangle ABC is circumscribed about a circle with radius #=10sqrt3# units. Notice that the sides of the triangle are tangent to the circle, we draw a line segment from the center O to the vertex A, we also draw the radius OD. OA bisects angle A to two 30 degrees parts. And OD is perpendicular to AC.
Triangle OAD is 30-60-90 triangle, the sides have a ratio of:
#1 : sqrt3 : 2# , hence: AD #=10sqrt3*sqrt3=30# units. Since OD bisects AC, we conclude that each side of the triangle ABC is 60 units therefore the perimeter is #3*60=180# units.