Finding the perimeter of an equilateral triangle using the Law of Sines - Solution

Question: 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?
Geometry question.

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Answered By:

Charles W.
Middle School Math Teacher who Loves Seeing Students Succeed

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Written Explanation:

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)/ACsimplifying the equation to find line segment AC...(sin 30) * (AC)= (sin 120)*6(.5) * (AC) = (.866) * 6AC = 5.196/.5AC = 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) = 31.177I 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.

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