College geometry

Question

12. In an equilateral triangle, find the length of a radius if the length of a side is 12.

A. 4√ 3

B. 6√ 3

C. 8√ 3

D. 12

E. None of the above

Answer 1

well if the circle is inscribed, then upon connecting the center to each triangle vertex produces

right triangles with legs R and 6 where R is the radius. Moreover, the angle at the tip is 30.

However, that would force the radius to be 6/sqrt(3) = 6*sqrt(3)/3 = 2*sqrt(3) which is not one of the choices.

Circumscribing the circle about the triangle, the right triangles created by connecting center to triangle

vertices, creates a height of 2*Sqrt(3) and base leg 6. Pythagorean then says the radius must be

sqrt( 12 + 36) = sqrt(48) = 4*sqrt(3) which is option A