If the centroid of a triangle formed by (7 , x) , (y , -6) and (9 , 10) is (6 , 3) , then the values of x and y are respectively

Options

A. 5, 2
B. 2 , 5
C. 1, 0
D. 0,0

Answer

To find the values of \( x \) and \( y \) for the triangle with vertices at \( (7, x) \), \( (y, -6) \), and \( (9, 10) \) with a centroid at \( (6, 3) \), we can use the formula for the centroid of a triangle. 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:

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

### Step 1: Set up the equations for the x-coordinates.

Using the coordinates of the vertices:
- \( x_1 = 7 \)
- \( x_2 = y \)
- \( x_3 = 9 \)

The x-coordinate of the centroid is given as 6:

\[ \frac{7 + y + 9}{3} = 6 \]

### Step 2: Solve for \( y \).

Multiply both sides by 3:

\[ 7 + y + 9 = 18 \]

Combine like terms:

\[ y + 16 = 18 \]

Subtract 16 from both sides:

\[ y = 2 \]

### Step 3: Set up the equations for the y-coordinates.

Using the coordinates of the vertices:
- \( y_1 = x \)
- \( y_2 = -6 \)
- \( y_3 = 10 \)

The y-coordinate of the centroid is given as 3:

\[ \frac{x + (-6) + 10}{3} = 3 \]

### Step 4: Solve for \( x \).

Multiply both sides by 3:

\[ x - 6 + 10 = 9 \]

Combine like terms:

\[ x + 4 = 9 \]

Subtract 4 from both sides:

\[ x = 5 \]

### Final Answer

The values of \( x \) and \( y \) are respectively:

\[ x = 5, \quad y = 2 \]

---