Three consecutive vertices of a parallelogram ABCD are A(3,-1,2) B, (1,2,-4) and C(-1,1,2), the fourth vertex D is

Options

A. (-1,2,8)
B. (1,-2,8)
C. (1,2,8)
D. (1,2,-8)

Answer

To find the fourth vertex \( D \) of the parallelogram \( ABCD \) given the vertices \( A(3, -1, 2) \), \( B(1, 2, -4) \), and \( C(-1, 1, 2) \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-step Solution:

1. **Identify the Coordinates of the Given Points:**
- \( A(3, -1, 2) \)
- \( B(1, 2, -4) \)
- \( C(-1, 1, 2) \)

2. **Let the Coordinates of Point D be \( D(x, y, z) \).**

3. **Find the Midpoint of Diagonal AC:**
The midpoint \( M_{AC} \) of diagonal \( AC \) can be calculated using the midpoint formula:
\[ M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting the coordinates of points \( A \) and \( C \):
\[ M_{AC} = \left( \frac{3 + (-1)}{2}, \frac{-1 + 1}{2}, \frac{2 + 2}{2} \right) = \left( \frac{2}{2}, \frac{0}{2}, \frac{4}{2} \right) = (1, 0, 2) \]

4. **Set the Midpoint of Diagonal BD Equal to Midpoint AC:**
The midpoint \( M_{BD} \) of diagonal \( BD \) is given by:
\[ M_{BD} = \left( \frac{1 + x}{2}, \frac{2 + y}{2}, \frac{-4 + z}{2} \right) \]
Since \( M_{AC} = M_{BD} \), we can set the coordinates equal to each other:
\[ \frac{1 + x}{2} = 1 \quad (1) \]
\[ \frac{2 + y}{2} = 0 \quad (2) \]
\[ \frac{-4 + z}{2} = 2 \quad (3) \]

5. **Solve the Equations:**
- From equation (1):
\[ 1 + x = 2 \implies x = 1 \]
- From equation (2):
\[ 2 + y = 0 \implies y = -2 \]
- From equation (3):
\[ -4 + z = 4 \implies z = 8 \]

6. **Conclusion:**
The coordinates of point \( D \) are:
\[ D(1, -2, 8) \]

### Final Answer:
The fourth vertex \( D \) of the parallelogram is \( D(1, -2, 8) \).