To find the area of triangle ABC, we can use the formula for the area of a triangle given two sides and the included angle. Let's call the center of the circle O.
Step 1: Find the length of side AB
Since points A and B are on the circumference of the circle centered at O with a radius of 2, the length of side AB is equal to the diameter of the circle, which is twice the radius:
AB = 2 * 2 = 4
Step 2: Find the length of side AC
Similarly, since points A and C are on the circumference of the circle centered at O with a radius of 2, the length of side AC is also equal to the diameter of the circle, which is twice the radius:
AC = 2 * 2 = 4
Step 3: Use the Law of Cosines to find the length of side BC
In triangle ABC, we can use the Law of Cosines to find the length of side BC. The Law of Cosines states that for any triangle with sides a, b, and c, and angle C (opposite to side c), the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, side a = AB = 4, side b = AC = 4, and angle C = angle ACB = 60 degrees. Let's calculate BC:
BC^2 = 4^2 + 4^2 - 2 * 4 * 4 * cos(60)
BC^2 = 16 + 16 - 32 * 0.5
BC^2 = 32 - 16
BC^2 = 16
BC = sqrt(16)
BC = 4
Step 4: Calculate the area of triangle ABC
Now that we have the lengths of all three sides of triangle ABC, we can use the formula for the area of a triangle given two sides and the included angle:
Area = (1/2) * AB * AC * sin(angle BAC)
In this case, AB = 4, AC = 4, and angle BAC = 45 degrees. Let's calculate the area:
Area = (1/2) * 4 * 4 * sin(45)
Area = 8 * sqrt(2) / 2
Area = 4 * sqrt(2)
Therefore, the area of triangle ABC is 4 * sqrt(2) square units.