The co-ordinates of two vertices of `Delta ABC` are A(-5,7,3) and B(7,-6,-1). The co-ordinates of its centroid are (1,1,1). Find the co-ordinates of vertex C.

Explanation

To find the coordinates of vertex C of triangle ABC given the coordinates of vertices A and B and the centroid, we can use the formula for the centroid of a triangle. The centroid (G) of a triangle with vertices A(x1, y1, z1), B(x2, y2, z2), and C(x3, y3, z3) is given by:

\[ G = \left( \frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}, \frac{z1 + z2 + z3}{3} \right) \]

### Step 1: Identify the coordinates of A and B
The coordinates of the vertices are:
- A(-5, 7, 3)
- B(7, -6, -1)

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

### Step 2: Set up the equations for the centroid
The coordinates of the centroid G are given as (1, 1, 1). Therefore, we can set up the following equations based on the centroid formula:

1. For the x-coordinate:
\[ \frac{-5 + 7 + x}{3} = 1 \]

2. For the y-coordinate:
\[ \frac{7 - 6 + y}{3} = 1 \]

3. For the z-coordinate:
\[ \frac{3 - 1 + z}{3} = 1 \]

### Step 3: Solve for x
From the first equation:
\[ \frac{-5 + 7 + x}{3} = 1 \]
Multiply both sides by 3:
\[ -5 + 7 + x = 3 \]
Simplify:
\[ 2 + x = 3 \]
Thus,
\[ x = 3 - 2 = 1 \]

### Step 4: Solve for y
From the second equation:
\[ \frac{7 - 6 + y}{3} = 1 \]
Multiply both sides by 3:
\[ 7 - 6 + y = 3 \]
Simplify:
\[ 1 + y = 3 \]
Thus,
\[ y = 3 - 1 = 2 \]

### Step 5: Solve for z
From the third equation:
\[ \frac{3 - 1 + z}{3} = 1 \]
Multiply both sides by 3:
\[ 3 - 1 + z = 3 \]
Simplify:
\[ 2 + z = 3 \]
Thus,
\[ z = 3 - 2 = 1 \]

### Conclusion
The coordinates of vertex C are:
\[ C(1, 2, 1) \]