A, B, and C are points on Circle O such that Triangle ABC is equilateral. If the radius of Circle O is 6, what is the perimeter of Triangle ABC?

Question

Geometry question.

Answer 1

Drop a vertical line down to point D like this:

ΔCOD is a 30-60-90 triangle so side OD is 3 and side CD is 3√3. That makes segment CB 6√3 and since that is one side of ΔABC, the perimeter is 18√3

Answer 2

Let's try to find AC using our knowledge that the radius of circle O is 6. Then we can use the value of AC to find the perimeter of the equilateral triangle

- Given the radius of circle is 6, this gives us the sides, and indirect interior angles of all of the interior triangles in the circumscribed triangle.

- OC = 6, OA = 6, AC = ?

- The angle of OCA is exactly half of ACB, which would be 60/2 = 30 degrees. The same would be true for angle OAC. Leaving the angle AOC to be the difference between the 2 congruent angles

- Angle OCA = 30 degrees, Angle OAC = 30 degrees,

Angle COA =[180 - 2(30)] = 120 degrees.

Now using the Law of Sines, we can create a proportion of the data we have to find the length of side AC....

(sine 30)/(OC) = (sin 30)/(OA) = (sin 120)/(AC)

Now, replacing the lengths of each line segment represented we have...

(sin 30)/6 = (sin 30)/6 = (sin 120)/AC

simplifying the equation to find line segment AC...

(sin 30) * (AC)= (sin 120)*6

(.5) * (AC) = (.866) * 6

AC = 5.196/.5

AC = 10.392...which is one side of the triangle ABC, but the question is asking for the perimeter of this equilateral triangle or...

3(10.392) = 31.177

I used a trigonometry-based solution for this one, but I think it is the most direct route in finding the correct answer. In a geometry class you may not be taught this until relatively later in the year.

Please let me know if I can clarify my solution or explanation.