The third vertex of a triangle, whose two vertices are (-4,1) and (0, -3) and centroid is at the origin is

Options

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

Answer

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

### Step 1: Identify the given points and centroid
We have two vertices of the triangle:
- Vertex A: \( (-4, 1) \)
- Vertex B: \( (0, -3) \)
The centroid (G) is given to be at the origin, which is \( (0, 0) \).

### Step 2: Use the centroid formula
The formula for the centroid \( G(x, y) \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:
\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \]
Since the centroid is at the origin, we have:
\[ G_x = 0, \quad G_y = 0 \]

### Step 3: Set up equations for x-coordinates
Substituting the known values into the x-coordinate formula:
\[ 0 = \frac{-4 + 0 + x}{3} \]
Multiplying both sides by 3:
\[ 0 = -4 + x \]
Solving for \( x \):
\[ x = 4 \]

### Step 4: Set up equations for y-coordinates
Now, substituting the known values into the y-coordinate formula:
\[ 0 = \frac{1 + (-3) + y}{3} \]
Multiplying both sides by 3:
\[ 0 = 1 - 3 + y \]
This simplifies to:
\[ 0 = -2 + y \]
Solving for \( y \):
\[ y = 2 \]

### Step 5: Conclusion
Thus, the coordinates of the third vertex \( C \) are:
\[ C(4, 2) \]

### Summary
The third vertex of the triangle is \( (4, 2) \).

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