If the three vertices of a parallelogram ABCD are
A(3, -1, 5), B(1, -2, -4) and C(0, 3, 0), then the
coordinates of fourth vertex is

Options

A. (2, 4, 9)
B. (2, -4, -9)
C. (0, 4, 9)
D. (0, -4, -9)

Answer

To find the coordinates of the fourth vertex \( D \) of the parallelogram \( ABCD \) given the vertices \( A(3, -1, 5) \), \( B(1, -2, -4) \), and \( C(0, 3, 0) \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Coordinates of Points A, B, and C:**
- \( A(3, -1, 5) \)
- \( B(1, -2, -4) \)
- \( C(0, 3, 0) \)

2. **Find the Midpoint \( O \) of Diagonal \( AC \):**
The midpoint \( O \) of segment \( AC \) can be calculated using the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting the coordinates of \( A \) and \( C \):
\[ O = \left( \frac{3 + 0}{2}, \frac{-1 + 3}{2}, \frac{5 + 0}{2} \right) = \left( \frac{3}{2}, 1, \frac{5}{2} \right) \]

3. **Express the Coordinates of Point D:**
Let the coordinates of point \( D \) be \( D(x, y, z) \). Since \( O \) is also the midpoint of diagonal \( BD \), we can write:
\[ O = \left( \frac{1 + x}{2}, \frac{-2 + y}{2}, \frac{-4 + z}{2} \right) \]

4. **Set Up the Equations:**
Since both expressions for \( O \) are equal, we can equate the corresponding components:
- For the x-coordinates:
\[ \frac{1 + x}{2} = \frac{3}{2} \]
- For the y-coordinates:
\[ \frac{-2 + y}{2} = 1 \]
- For the z-coordinates:
\[ \frac{-4 + z}{2} = \frac{5}{2} \]

5. **Solve for x, y, and z:**
- From the x-coordinate equation:
\[ 1 + x = 3 \implies x = 3 - 1 = 2 \]
- From the y-coordinate equation:
\[ -2 + y = 2 \implies y = 2 + 2 = 4 \]
- From the z-coordinate equation:
\[ -4 + z = 5 \implies z = 5 + 4 = 9 \]

6. **Conclusion:**
The coordinates of the fourth vertex \( D \) are:
\[ D(2, 4, 9) \]

### Final Answer:
The coordinates of the fourth vertex \( D \) are \( (2, 4, 9) \).

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