Two vertices of a `triangle`ABC are B (- 3,1) and C (0,-2) and its centroid is at the origin. The third vertex A is:

Options

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

Answer

To find the coordinates of the third vertex \( A \) of triangle \( ABC \) given the coordinates of vertices \( B \) and \( C \) and the centroid at the origin, we can follow these steps:

### Step 1: Understand the Centroid Formula
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by the formula:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Since the centroid is at the origin, we have:
\[ G(0, 0) \]

### Step 2: Set Up the Equations
Given the coordinates of \( B \) and \( C \):
- \( B(-3, 1) \) implies \( x_2 = -3 \) and \( y_2 = 1 \)
- \( C(0, -2) \) implies \( x_3 = 0 \) and \( y_3 = -2 \)

We can substitute these values into the centroid formula:
\[ \frac{x_1 - 3 + 0}{3} = 0 \quad \text{(for the x-coordinate)} \]
\[ \frac{y_1 + 1 - 2}{3} = 0 \quad \text{(for the y-coordinate)} \]

### Step 3: Solve for \( x_1 \)
From the x-coordinate equation:
\[ \frac{x_1 - 3 + 0}{3} = 0 \]
Multiplying both sides by 3:
\[ x_1 - 3 = 0 \]
Adding 3 to both sides:
\[ x_1 = 3 \]

### Step 4: Solve for \( y_1 \)
From the y-coordinate equation:
\[ \frac{y_1 + 1 - 2}{3} = 0 \]
Multiplying both sides by 3:
\[ y_1 + 1 - 2 = 0 \]
Simplifying:
\[ y_1 - 1 = 0 \]
Adding 1 to both sides:
\[ y_1 = 1 \]

### Step 5: Conclusion
Thus, the coordinates of vertex \( A \) are:
\[ A(3, 1) \]