To find the coordinates of the centroid of a triangle, we can use the formula:

Centroid = (x1 + x2 + x3)/3, (y1 + y2 + y3)/3, (z1 + z2 + z3)/3

Given the coordinates of the vertices of the triangle as (7, 6, 4), (5, 4, 6), and (9, 5, 8), we can substitute the values into the formula to find the centroid.

Calculating the centroid using the formula:

x-coordinate of centroid = (7 + 5 + 9)/3 = 21/3 = 7
y-coordinate of centroid = (6 + 4 + 5)/3 = 15/3 = 5
z-coordinate of centroid = (4 + 6 + 8)/3 = 18/3 = 6

Therefore, the coordinates of the centroid of the triangle are (7, 5, 6).

Explanation:

The centroid of a triangle is the point of intersection of the medians of the triangle. A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side.

To find the centroid, we first need to find the midpoint of each side of the triangle. The midpoint of a line segment with coordinates (x1, y1, z1) and (x2, y2, z2) can be found using the formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)

Using this formula, we can find the midpoints of the sides of the triangle:

Midpoint of side 1 = ((7 + 5)/2, (6 + 4)/2, (4 + 6)/2) = (6, 5, 5)
Midpoint of side 2 = ((7 + 9)/2, (6 + 5)/2, (4 + 8)/2) = (8, 5.5, 6)
Midpoint of side 3 = ((5 + 9)/2, (4 + 5)/2, (6 + 8)/2) = (7, 4.5, 7)

Next, we need to find the coordinates of the centroid, which is the average of the coordinates of the midpoints of the sides:

x-coordinate of centroid = (6 + 8 + 7)/3 = 21/3 = 7
y-coordinate of centroid = (5 + 5.5 + 4.5)/3 = 15/3 = 5
z-coordinate of centroid = (5 + 6 + 7)/3 = 18/3 = 6

Therefore, the coordinates of the centroid of the triangle are (7, 5, 6).