Two vertices of a triangle are (3, 4, 2) and (1, 3, 3) If the medians of the triangle intersect at (0, 3,-1), then the coordinates of the third vertex of the triangle are

Options

A. (3, 1, -8)
B. (4, 2, 8)
C. (0, 0, 0)
D. (-4, 2, -8)

Answer

To find the coordinates of the third vertex of the triangle given the other two vertices and the centroid, we can follow these steps:

### Step 1: Identify the vertices and centroid
Let the vertices of the triangle be:
- Vertex A: \( A(3, 4, 2) \)
- Vertex B: \( B(1, 3, 3) \)
- Vertex C: \( C(x, y, z) \) (the coordinates we need to find)
- Centroid G: \( G(0, 3, -1) \)

### Step 2: Use the formula for the centroid
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

### Step 3: Set up equations for each coordinate
From the centroid formula, we can set up the following equations based on the given coordinates:
1. For the x-coordinate:
\[ 0 = \frac{3 + 1 + x}{3} \]
2. For the y-coordinate:
\[ 3 = \frac{4 + 3 + y}{3} \]
3. For the z-coordinate:
\[ -1 = \frac{2 + 3 + z}{3} \]

### Step 4: Solve for x
From the first equation:
\[ 0 = \frac{4 + x}{3} \]
Multiplying both sides by 3:
\[ 0 = 4 + x \implies x = -4 \]

### Step 5: Solve for y
From the second equation:
\[ 3 = \frac{7 + y}{3} \]
Multiplying both sides by 3:
\[ 9 = 7 + y \implies y = 9 - 7 = 2 \]

### Step 6: Solve for z
From the third equation:
\[ -1 = \frac{5 + z}{3} \]
Multiplying both sides by 3:
\[ -3 = 5 + z \implies z = -3 - 5 = -8 \]

### Step 7: Write the coordinates of the third vertex
Thus, the coordinates of the third vertex \( C \) are:
\[ C(-4, 2, -8) \]

### Final Answer
The coordinates of the third vertex of the triangle are \( (-4, 2, -8) \).

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