Two vertices of `Delta ABC` are A(-1, 4) and B(5, 2) and its centroid is G(0, -3). Then, the coordinates of C are

Options

A. (4, 3)
B. (4, 15)
C. (-4, -15)
D. (-15, -4)

Explanation

To find the coordinates of vertex C of triangle ABC given the coordinates of vertices A and B and the centroid G, we can use the formula for the centroid of a triangle. The centroid G of a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3) is given by:

\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

Given:
- A(-1, 4)
- B(5, 2)
- G(0, -3)

Let the coordinates of C be (x, y).

### Step 1: Set up the equations for the centroid
Using the formula for the centroid, we can set up the following equations:

\[ G_x = \frac{x_1 + x_2 + x_3}{3} \quad \text{and} \quad G_y = \frac{y_1 + y_2 + y_3}{3} \]

Substituting the known values:

\[ 0 = \frac{-1 + 5 + x}{3} \quad \text{(1)} \]
\[ -3 = \frac{4 + 2 + y}{3} \quad \text{(2)} \]

### Step 2: Solve for x using equation (1)
Multiply both sides of equation (1) by 3:

\[ 0 = -1 + 5 + x \]

Simplifying this gives:

\[ 0 = 4 + x \]

So,

\[ x = -4 \]

### Step 3: Solve for y using equation (2)
Multiply both sides of equation (2) by 3:

\[ -9 = 4 + 2 + y \]

Simplifying this gives:

\[ -9 = 6 + y \]

So,

\[ y = -9 - 6 = -15 \]

### Step 4: Write the coordinates of C
Thus, the coordinates of point C are:

\[ C(-4, -15) \]

### Final Answer
The coordinates of C are (-4, -15).

---