Question

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

Answer 1

The perimeter of the triangle is 180 units.

Explanation:

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.