The co-ordinates of two vertices of `Delta ABC` are A(8,-9,8) and B(1,2,3). The medians of the triangle meet at the point (5,-2,4). Find the co-ordinates of the vertex C.

Explanation

To find the coordinates of vertex C of triangle ABC given the coordinates of vertices A and B and the centroid G, we can use the properties of the centroid. The centroid (G) of a triangle is the average of the coordinates of its vertices.

### Step-by-step Solution:

1. **Identify the coordinates of the given points:**
- A(8, -9, 8)
- B(1, 2, 3)
- G(5, -2, 4) (the centroid)

2. **Let the coordinates of vertex C be (x, y, z).**

3. **Use the formula for the centroid:**
The coordinates of the centroid G can be calculated using the formula:
\[ G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}, \frac{z_A + z_B + z_C}{3} \right) \]
where \( (x_A, y_A, z_A) \) are the coordinates of A, \( (x_B, y_B, z_B) \) are the coordinates of B, and \( (x_C, y_C, z_C) \) are the coordinates of C.

4. **Set up the equations for each coordinate:**

- For the x-coordinate:
\[ \frac{8 + 1 + x}{3} = 5 \]
Multiply both sides by 3:
\[ 8 + 1 + x = 15 \]
Simplifying gives:
\[ x = 15 - 9 = 6 \]

- For the y-coordinate:
\[ \frac{-9 + 2 + y}{3} = -2 \]
Multiply both sides by 3:
\[ -9 + 2 + y = -6 \]
Simplifying gives:
\[ y = -6 + 7 = 1 \]

- For the z-coordinate:
\[ \frac{8 + 3 + z}{3} = 4 \]
Multiply both sides by 3:
\[ 8 + 3 + z = 12 \]
Simplifying gives:
\[ z = 12 - 11 = 1 \]

5. **Combine the results:**
The coordinates of vertex C are (6, 1, 1).

### Final Answer:
The coordinates of vertex C are **C(6, 1, 1)**.