Two vertices of a triangle are at (- 3, 1) and (0,2) and the centroid is at the origin. Find its third vertex.

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 first vertex \( A \) be at \( (-3, 1) \).
- Let the second vertex \( B \) be at \( (0, 2) \).
- Let the third vertex \( C \) be at \( (a, b) \) (this is what we need to find).
- The centroid \( G \) is at the origin \( (0, 0) \).

2. **Use the Centroid Formula:**
The formula for the centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \]
Here, since the centroid is at the origin, both \( G_x \) and \( G_y \) are equal to 0.

3. **Set Up the Equations:**
For the x-coordinates:
\[ 0 = \frac{-3 + 0 + a}{3} \]
For the y-coordinates:
\[ 0 = \frac{1 + 2 + b}{3} \]

4. **Solve for \( a \):**
From the first equation:
\[ 0 = \frac{-3 + a}{3} \]
Multiplying both sides by 3 gives:
\[ 0 = -3 + a \implies a = 3 \]

5. **Solve for \( b \):**
From the second equation:
\[ 0 = \frac{1 + 2 + b}{3} \]
Multiplying both sides by 3 gives:
\[ 0 = 3 + b \implies b = -3 \]

6. **Conclusion:**
The coordinates of the third vertex \( C \) are \( (a, b) = (3, -3) \).

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

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