Final answer:
The coordinates of the centroid of the triangle are (3.33, 4) and the circumradius is approximately 0.56.
Step-by-step explanation:
To find the centroid of a triangle, we need to find the average of the x-coordinates and the average of the y-coordinates of the three vertices. Let's calculate:
x-coordinate of centroid = (4 + 0 + 6)/3 = 10/3 = 3.33 (rounded to two decimal places)
y-coordinate of centroid = (6 + 4 + 2)/3 = 12/3 = 4
Therefore, the coordinates of the centroid are (3.33, 4).
To find the circumradius of a triangle, we can use the formula:
Circumradius = (a * b * c) / (4 * Area)
Where a, b, and c are the lengths of the sides of the triangle, and Area is the area of the triangle. In this case, we can use the distance formula to find the lengths of the sides:
Side AB = sqrt((0 - 4)^2 + (4 - 6)^2) = sqrt(16 + 4) = sqrt(20)
Side BC = sqrt((6 - 0)^2 + (2 - 4)^2) = sqrt(36 + 4) = sqrt(40)
Side AC = sqrt((4 - 6)^2 + (6 - 2)^2) = sqrt(4 + 16) = sqrt(20)
Now, let's find the area of the triangle using the formula:
Area = 1/2 * base * height
The base is the length of Side BC, and the height can be found by drawing an altitude from vertex A to Side BC:
Height = sqrt((6 - 4)^2 + (2 - 4)^2) = sqrt(4 + 4) = sqrt(8)
Area = 1/2 * sqrt(40) * sqrt(8) = sqrt(320)
Finally, let's calculate the circumradius:
Circumradius = (sqrt(20) * sqrt(40) * sqrt(20)) / (4 * sqrt(320)) = sqrt(160) / 8 = 2sqrt(5) / 8 = sqrt(5) / 4 = 0.56 (rounded to two decimal places)
Therefore, the coordinates of the centroid of the triangle are (3.33, 4) and the circumradius is approximately 0.56.