- Centroid Calculation:
The centroid of a triangle is the point where the three medians intersect. The medians are the line segments drawn from each vertex of the triangle to the midpoint of the opposite side. The coordinates of the centroid can be calculated by finding the average of the x-coordinates, y-coordinates, and z-coordinates of the vertices of the triangle.
- Finding the Midpoints:
First, we need to find the midpoints of the sides of the triangle. The midpoint of a line segment with endpoints (x1, y1, z1) and (x2, y2, z2) is given by the formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2, (z1 + z2) / 2)
- Midpoint of side 1:
Midpoint1 = ((7 + 15) / 2, (-2 - 10) / 2, (5 + 16) / 2) = (11, -6, 10.5)
- Midpoint of side 2:
Midpoint2 = ((15 + 2) / 2, (-10 + 6) / 2, (16 - 4) / 2) = (8.5, -2, 6)
- Midpoint of side 3:
Midpoint3 = ((2 + 7) / 2, (6 - 2) / 2, (-4 + 5) / 2) = (4.5, 2, 0.5)
- Calculating the Centroid:
Now, we can find the centroid by taking the average of the x-coordinates, y-coordinates, and z-coordinates of the midpoints:
Centroid = ((11 + 8.5 + 4.5) / 3, (-6 - 2 + 2) / 3, (10.5 + 6 + 0.5) / 3)
Centroid = (8, -2, 5)
Therefore, the centroid of the triangle with vertices at (7,-2,5), (15,-10,16), and (2,6,-4) is (8, -2, 5).