The centroid of a triangle formed by (7, p), (q, -6), (9, 10) is (6, 3). Then p + q

Options

A. 6
B. 5
C. 7
D. 8

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) using the coordinates of the triangle's vertices and the given centroid. The vertices of the triangle are \( (7, p) \), \( (q, -6) \), and \( (9, 10) \), and the centroid is given as \( (6, 3) \).

### Step-by-Step Solution:

1. **Understand the Centroid Formula**:
The centroid \( (G) \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \]
where \( G_x \) and \( G_y \) are the x and y coordinates of the centroid, respectively.

2. **Set Up the Equations**:
For our triangle:
- \( (x_1, y_1) = (7, p) \)
- \( (x_2, y_2) = (q, -6) \)
- \( (x_3, y_3) = (9, 10) \)

The centroid is \( (6, 3) \). Therefore, we can set up the following equations based on the centroid formula:

- For the x-coordinate:
\[ \frac{7 + q + 9}{3} = 6 \]
- For the y-coordinate:
\[ \frac{p - 6 + 10}{3} = 3 \]

3. **Solve the x-coordinate Equation**:
Multiply both sides of the x-coordinate equation by 3:
\[ 7 + q + 9 = 18 \]
Simplifying this gives:
\[ q + 16 = 18 \]
Therefore:
\[ q = 18 - 16 = 2 \]

4. **Solve the y-coordinate Equation**:
Multiply both sides of the y-coordinate equation by 3:
\[ p - 6 + 10 = 9 \]
Simplifying this gives:
\[ p + 4 = 9 \]
Therefore:
\[ p = 9 - 4 = 5 \]

5. **Calculate \( p + q \)**:
Now that we have \( p = 5 \) and \( q = 2 \):
\[ p + q = 5 + 2 = 7 \]

### Final Answer:
Thus, the value of \( p + q \) is \( \boxed{7} \).