The co-ordinates of two vertices of `Delta ABC` are A(3,2,-4) and B(-2,3,-1). If its centroid is (3,1,0), then 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 G, we can follow these steps:

### Step 1: Understand the formula for the centroid
The centroid (G) of a triangle with vertices A(x1, y1, z1), B(x2, y2, z2), and C(x3, y3, z3) is given by the formula:
\[ G = \left( \frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}, \frac{z1 + z2 + z3}{3} \right) \]

### Step 2: Assign the known coordinates
We have:
- A(3, 2, -4)
- B(-2, 3, -1)
- G(3, 1, 0)
- Let C's coordinates be (x, y, z)

### Step 3: Set up the equations
Using the centroid formula, we can set up the following equations based on the coordinates of G:
1. For the x-coordinates:
\[ 3 = \frac{3 + (-2) + x}{3} \]
2. For the y-coordinates:
\[ 1 = \frac{2 + 3 + y}{3} \]
3. For the z-coordinates:
\[ 0 = \frac{-4 + (-1) + z}{3} \]

### Step 4: Solve the equations

**Equation 1 (x-coordinate):**
\[ 3 = \frac{3 - 2 + x}{3} \]
Multiply both sides by 3:
\[ 9 = 1 + x \]
Subtract 1 from both sides:
\[ x = 8 \]

**Equation 2 (y-coordinate):**
\[ 1 = \frac{2 + 3 + y}{3} \]
Multiply both sides by 3:
\[ 3 = 5 + y \]
Subtract 5 from both sides:
\[ y = -2 \]

**Equation 3 (z-coordinate):**
\[ 0 = \frac{-4 - 1 + z}{3} \]
Multiply both sides by 3:
\[ 0 = -5 + z \]
Add 5 to both sides:
\[ z = 5 \]

### Step 5: Write the coordinates of vertex C
Thus, the coordinates of vertex C are:
\[ C(8, -2, 5) \]

### Final Answer
The coordinates of vertex C are (8, -2, 5).

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