An equilateral triangle with one side 6cm is inscribed in a circle with its three vertices on the circumference. What is the diameter of the circle?

(In reply to re: Trigonometry? (Solution) by Ender)

I thought it was obvious what I was doing, but I guess not.

Like Levik and Ender, I started with a 30-60-90 triangle, although a larger one than Levik's. I used CMA. We were given that CA was 6 and it was obvious that AM was 3. That meant that CM was 3*(SQRT[3]).

Drawing OA gives us tiangle AMO which is similar to (has the same angles as) triangle CMA, which means that just as CA is twice AM, AO is twice OM. Since OC and OA are both radii, they are equal. CM = OC + OM = OA + OM = (2 * OM) + OM = 3 * OM. But CM = 3 * (SQRT[3]), so OM = SQRT(3) and OA = 2 * (SQRT[3]), so the diameter = 4 * (SQRT[3]).

When I did it in my head, it was all so obvious, but working it out I need to include all of Ender's details and more. Go figure.

Posted by TomM on 2002-06-04 04:16:07