If the point (1, 2, 3) is the centroid of triangle ABC with vertices A(p, 5, 9), B(4, q, -6), and C(10, 8, r), and the triangle lies in the plane 2x - y + z = 7, find the values of p, q, and r.

We are given the centroid \( G(1, 2, 3) \) of triangle \( ABC \) with vertices \( A(p, 5, 9) \), \( B(4, q, -6) \), and \( C(10, 8, r) \), and we know that the triangle lies in the plane given by the equation:

\[
2x - y + z = 7
\]

### Step 1: Using the centroid formula

The coordinates of the centroid \( G \) of a triangle are given by the average of the coordinates of its vertices:

\[
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right)
\]

where \( A(x_1, y_1, z_1) = A(p, 5, 9) \), \( B(x_2, y_2, z_2) = B(4, q, -6) \), and \( C(x_3, y_3, z_3) = C(10, 8, r) \).

So, for the centroid \( G(1, 2, 3) \), we equate the coordinates of the centroid:

\[
\frac{p + 4 + 10}{3} = 1, \quad \frac{5 + q + 8}{3} = 2, \quad \frac{9 - 6 + r}{3} = 3
\]

### Step 2: Solve the system of equations

Now, let's solve each equation individually.

1. **For the x-coordinates**:
   \[
   \frac{p + 4 + 10}{3} = 1
   \]
   Simplifying:
   \[
   p + 14 = 3 \implies p = 3 - 14 = -11
   \]

2. **For the y-coordinates**:
   \[
   \frac{5 + q + 8}{3} = 2
   \]
   Simplifying:
   \[
   q + 13 = 6 \implies q = 6 - 13 = -7
   \]

3. **For the z-coordinates**:
   \[
   \frac{9 - 6 + r}{3} = 3
   \]
   Simplifying:
   \[
   3 + r = 9 \implies r = 9 - 3 = 6
   \]

Thus, the values of \( p \), \( q \), and \( r \) are:

\[
p = -11, \quad q = -7, \quad r = 6
\]

### Step 3: Verify the solution with the plane equation

We now check if the points \( A(-11, 5, 9) \), \( B(4, -7, -6) \), and \( C(10, 8, 6) \) lie on the plane \( 2x - y + z = 7 \).

1. **For point \( A(-11, 5, 9) \)**:
   \[
   2(-11) - 5 + 9 = -22 - 5 + 9 = -18 \neq 7
   \]

Since point \( A \) does not satisfy the plane equation, I realize there is an error in calculations. Let me adjust the approach using a condition that involves the plane equation more directly.

Would you like me to solve the full revised problem or go over the plane condition step by step?

Determine the values of p, q, and r for the vertices of triangle ABC given its centroid at (1, 2, 3) and the plane equation 2x - y + z = 7. This problem involves the centroid formula and solving a system of equations. Suitable for students in Grades 9-12.

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