Finding the Centroid of a Triangle:

To find the centroid of a triangle, we need to calculate the average of the coordinates of its three vertices. The centroid is the point where the three medians of a triangle intersect. The medians are the line segments drawn from each vertex of the triangle to the midpoint of the opposite side.

Step 1: Find the Midpoints of the Triangle's Sides:
To find the midpoints of the triangle's sides, we can use the midpoint formula, which states that the midpoint of a line segment with endpoints (x1, y1, z1) and (x2, y2, z2) is ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2).

Given the vertices of the triangle are (3, 6, 5), (6, 2, 7), and (3, 1, 9), we can find the midpoints of the sides as follows:

Midpoint of the side connecting (3, 6, 5) and (6, 2, 7):
((3+6)/2, (6+2)/2, (5+7)/2) = (4.5, 4, 6)

Midpoint of the side connecting (3, 6, 5) and (3, 1, 9):
((3+3)/2, (6+1)/2, (5+9)/2) = (3, 3.5, 7)

Midpoint of the side connecting (6, 2, 7) and (3, 1, 9):
((6+3)/2, (2+1)/2, (7+9)/2) = (4.5, 1.5, 8)

Step 2: Calculate the Centroid:
To find the centroid, we calculate the average of the coordinates of the midpoints obtained in Step 1.

Centroid = ((4.5+4.5+3)/3, (4+3.5+1.5)/3, (6+7+8)/3)
= (4, 3.33, 7)

Therefore, the centroid of the triangle with vertices (3, 6, 5), (6, 2, 7), and (3, 1, 9) is approximately (4, 3.33, 7).

Summary:
- To find the centroid of a triangle, calculate the average of the coordinates of its three vertices.
- Find the midpoints of the triangle's sides using the midpoint formula.
- Calculate the centroid by averaging the coordinates of the midpoints.
- The centroid is the point where the three medians of a triangle intersect.
- In this case, the centroid of the given triangle is approximately (4, 3.33, 7).