If `G(r, (-4)/(3), (1)/(3))` is centroid of the triangle having vertices `A(5, 1, p), B(1,q, p), C(1, -2, 3)`, then

Options

A. `p=-1, q=-3, r=(7)/(3)`
B. `p=1, q=-3, r=(7)/(3)`
C. `p=-1, q=3, r=(7)/(3)`
D. `p=1, q=3, r=(7)/(3)`

Answer

To solve the problem, we will find the values of \( P \), \( Q \), and \( R \) given that \( G(r, -\frac{4}{3}, \frac{1}{3}) \) is the centroid of the triangle with vertices \( A(5, 1, p) \), \( B(1, q, p) \), and \( C(1, -2, 3) \).

### Step 1: Write the formula for the centroid of a triangle
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

### Step 2: Substitute the coordinates of the vertices into the centroid formula
For our triangle:
- \( A(5, 1, p) \)
- \( B(1, q, p) \)
- \( C(1, -2, 3) \)

The coordinates of the centroid \( G \) will be:
\[ G = \left( \frac{5 + 1 + 1}{3}, \frac{1 + q - 2}{3}, \frac{p + p + 3}{3} \right) \]

### Step 3: Simplify the expressions for the centroid
Calculating each coordinate:
1. **X-coordinate**:
\[ \frac{5 + 1 + 1}{3} = \frac{7}{3} \]
2. **Y-coordinate**:
\[ \frac{1 + q - 2}{3} = \frac{q - 1}{3} \]
3. **Z-coordinate**:
\[ \frac{p + p + 3}{3} = \frac{2p + 3}{3} \]

Thus, we have:
\[ G = \left( \frac{7}{3}, \frac{q - 1}{3}, \frac{2p + 3}{3} \right) \]

### Step 4: Set the coordinates of the centroid equal to the given coordinates
We know that:
\[ G = \left( r, -\frac{4}{3}, \frac{1}{3} \right) \]

From this, we can set up the following equations:
1. For the x-coordinate:
\[ r = \frac{7}{3} \]
2. For the y-coordinate:
\[ -\frac{4}{3} = \frac{q - 1}{3} \]
3. For the z-coordinate:
\[ \frac{1}{3} = \frac{2p + 3}{3} \]

### Step 5: Solve for \( r \), \( q \), and \( p \)
1. From the first equation, we have:
\[ r = \frac{7}{3} \]

2. From the second equation, multiply both sides by 3:
\[ -4 = q - 1 \implies q = -4 + 1 = -3 \]

3. From the third equation, multiply both sides by 3:
\[ 1 = 2p + 3 \implies 2p = 1 - 3 = -2 \implies p = -1 \]

### Final Values
Thus, we have:
- \( P = -1 \)
- \( Q = -3 \)
- \( R = \frac{7}{3} \)

### Summary
The values are:
- \( P = -1 \)
- \( Q = -3 \)
- \( R = \frac{7}{3} \)