The third vertex of triangle whose centroid is origin and two vertex are (0,-2,5) and (-2,-2,-1) is

Options

A. (2,4,-4)
B. (2,-4,-4)
C. (-2,4,-4)
D. (-2,-4,-4)

Answer

To find the third vertex of the triangle whose centroid is at the origin (0, 0, 0) and two vertices are given as A(0, -2, 5) and B(-2, -2, -1), we can use the formula for the centroid of a triangle.

### Step-by-Step Solution:

1. **Understanding the Centroid Formula**:
The centroid (G) of a triangle with vertices A(x1, y1, z1), B(x2, y2, z2), and C(x3, y3, z3) is given by:
\[ G = \left( \frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}, \frac{z1 + z2 + z3}{3} \right) \]
Since the centroid is at the origin, we have:
\[ G = (0, 0, 0) \]

2. **Setting Up the Equations**:
Given the vertices:
- A(0, -2, 5) → (x1, y1, z1) = (0, -2, 5)
- B(-2, -2, -1) → (x2, y2, z2) = (-2, -2, -1)
- C(x3, y3, z3) → (unknown vertex)

We can set up the equations for the centroid:
\[ 0 = \frac{0 + (-2) + x3}{3} \quad (1) \]
\[ 0 = \frac{-2 + (-2) + y3}{3} \quad (2) \]
\[ 0 = \frac{5 + (-1) + z3}{3} \quad (3) \]

3. **Solving for x3**:
From equation (1):
\[ 0 = \frac{-2 + x3}{3} \]
Multiplying both sides by 3:
\[ 0 = -2 + x3 \]
Thus,
\[ x3 = 2 \]

4. **Solving for y3**:
From equation (2):
\[ 0 = \frac{-4 + y3}{3} \]
Multiplying both sides by 3:
\[ 0 = -4 + y3 \]
Thus,
\[ y3 = 4 \]

5. **Solving for z3**:
From equation (3):
\[ 0 = \frac{4 + z3}{3} \]
Multiplying both sides by 3:
\[ 0 = 4 + z3 \]
Thus,
\[ z3 = -4 \]

6. **Final Vertex**:
The coordinates of the third vertex C are:
\[ C(2, 4, -4) \]

### Conclusion:
The third vertex of the triangle is \( C(2, 4, -4) \).