Find the centroid of a kite with points A(1, 6, 1), B(1, 3, 2), C(3, 2, 7), D(4, 5, 3). Show your complete solution.

To find the centroid of a kite defined by the vertices A(1, 6, 1), B(1, 3, 2), C(3, 2, 7), and D(4, 5, 3), we’ll use the formula for the centroid (G) of a polygon, which is given as:

G = (x₁ + x₂ + x₃ + … + x_n)/n, (y₁ + y₂ + y₃ + … + y_n)/n, (z₁ + z₂ + z₃ + … + z_n)/n

Here, n is the number of vertices, which in this case is 4.

Now, we will calculate the coordinates for the centroid:

• For x-coordinates: 1 (from A) + 1 (from B) + 3 (from C) + 4 (from D) = 9
• For y-coordinates: 6 (from A) + 3 (from B) + 2 (from C) + 5 (from D) = 16
• For z-coordinates: 1 (from A) + 2 (from B) + 7 (from C) + 3 (from D) = 13

Now, we divide by the number of points (n = 4):

• x-coordinate: 9 / 4 = 2.25
• y-coordinate: 16 / 4 = 4
• z-coordinate: 13 / 4 = 3.25

Putting it all together, the centroid G of the kite is at the point:

G(2.25, 4, 3.25)

This is the final result for the centroid of the kite formed by the given points.