Find the third vertex of the triangle whose two vertices are (-3, 1) and (0, -2) and the centroid is the origin.

Options

A. (2,3)
B. `((-14)/(3),(14)/(3))`
C. (3,1)
D. (6,4)

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Points:**
- Let the two vertices of the triangle be \( A(-3, 1) \) and \( B(0, -2) \).
- Let the coordinates of the third vertex \( C \) be \( (x, y) \).

2. **Use the Centroid Formula:**
- The centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
- For our triangle, the coordinates of the centroid are given as the origin \( (0, 0) \).

3. **Set Up the Equations:**
- From the centroid formula, we have:
\[ \frac{-3 + 0 + x}{3} = 0 \quad \text{(1)} \]
\[ \frac{1 - 2 + y}{3} = 0 \quad \text{(2)} \]

4. **Solve for \( x \) from Equation (1):**
- Multiply both sides of Equation (1) by 3:
\[ -3 + 0 + x = 0 \]
- Rearranging gives:
\[ x = 3 \]

5. **Solve for \( y \) from Equation (2):**
- Multiply both sides of Equation (2) by 3:
\[ 1 - 2 + y = 0 \]
- Rearranging gives:
\[ y = 1 \]

6. **Conclusion:**
- Therefore, the coordinates of the third vertex \( C \) are \( (3, 1) \).

### Final Answer:
The third vertex of the triangle is \( C(3, 1) \).

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