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.

The centroid of a triangle is the average of the coordinates of its three vertices. Given the centroid \((1, 2, 3)\) and the vertices of the triangle \(A(p, 5, 9)\), \(B(4, q, -6)\), and \(C(10, 8, r)\), we can use the formula for the coordinates of the centroid:

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

We now equate the coordinates to find \(p\), \(q\), and \(r\).

### Step 1: Find \(p\), \(q\), and \(r\) using the centroid condition

From the \(x\)-coordinate:
\[
\frac{p + 4 + 10}{3} = 1 \quad \Rightarrow \quad p + 14 = 3 \quad \Rightarrow \quad p = -11
\]

From the \(y\)-coordinate:
\[
\frac{5 + q + 8}{3} = 2 \quad \Rightarrow \quad q + 13 = 6 \quad \Rightarrow \quad q = -7
\]

From the \(z\)-coordinate:
\[
\frac{9 - 6 + r}{3} = 3 \quad \Rightarrow \quad r + 3 = 9 \quad \Rightarrow \quad r = 6
\]

Thus, we have \(p = -11\), \(q = -7\), and \(r = 6\).

### Step 2: Verify the points lie on the plane \(2x - y + z = 7\)

For vertex \(A(-11, 5, 9)\):
\[
2(-11) - 5 + 9 = -22 - 5 + 9 = -18 \quad (\text{not on the plane})
\]

For vertex \(B(4, -7, -6)\):
\[
2(4) - (-7) + (-6) = 8 + 7 - 6 = 9 \quad (\text{not on the plane})
\]

For vertex \(C(10, 8, 6)\):
\[
2(10) - 8 + 6 = 20 - 8 + 6 = 18 \quad (\text{not on the plane})


This problem involves finding the unknown coordinates of vertices A, B, and C of a triangle given its centroid and the plane it lies on. It covers key concepts like the centroid formula in coordinate geometry and the plane equation 2x - y + z = 7. Suitable for high school students in Grades 10-12.

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