Find the value of (x, y), if centroid of the triangle with vertices (x, 0), (0, y) and (6, 3) is (3, 4).

Options

A. (3, 0)
B. (6, 6)
C. (3, 9)
D. (-6, 8)

Answer

To find the values of \( x \) and \( y \) such that the centroid of the triangle with vertices \( (x, 0) \), \( (0, y) \), and \( (6, 3) \) is \( (3, 4) \), we can follow these steps:

### Step 1: Use the formula for the centroid
The formula for the centroid \( (G_x, G_y) \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \]

### Step 2: Substitute the vertices into the formula
Here, the vertices are \( (x, 0) \), \( (0, y) \), and \( (6, 3) \). Thus, we can write:
\[ G_x = \frac{x + 0 + 6}{3}, \quad G_y = \frac{0 + y + 3}{3} \]

### Step 3: Set the centroid equal to the given coordinates
We know the centroid is \( (3, 4) \). Therefore, we can set up the equations:
\[ \frac{x + 6}{3} = 3 \quad \text{(1)} \]
\[ \frac{y + 3}{3} = 4 \quad \text{(2)} \]

### Step 4: Solve for \( x \) using equation (1)
To solve for \( x \), multiply both sides of equation (1) by 3:
\[ x + 6 = 9 \]
Now, subtract 6 from both sides:
\[ x = 9 - 6 = 3 \]

### Step 5: Solve for \( y \) using equation (2)
To solve for \( y \), multiply both sides of equation (2) by 3:
\[ y + 3 = 12 \]
Now, subtract 3 from both sides:
\[ y = 12 - 3 = 9 \]

### Conclusion
Thus, the values of \( x \) and \( y \) are:
\[ (x, y) = (3, 9) \]