If G(a,2,-1) is the centroid of the triangle with vertices P(1,3,2),Q(3,b,-4) and R(5,1,c), then find the values of a,b and c.

Answer

To find the values of \( a \), \( b \), and \( c \) given that \( G(a, 2, -1) \) is the centroid of the triangle with vertices \( P(1, 3, 2) \), \( Q(3, b, -4) \), and \( R(5, 1, c) \), we can use the formula for the centroid of a triangle in 3D space.

### Step 1: Use the centroid formula for the x-coordinate
The x-coordinate of the centroid \( G \) is given by:
\[ x_G = \frac{x_1 + x_2 + x_3}{3} \]
Substituting the coordinates of points \( P \), \( Q \), and \( R \):
\[ a = \frac{1 + 3 + 5}{3} \]
Calculating this gives:
\[ a = \frac{9}{3} = 3 \]

### Step 2: Use the centroid formula for the y-coordinate
The y-coordinate of the centroid \( G \) is given by:
\[ y_G = \frac{y_1 + y_2 + y_3}{3} \]
Substituting the coordinates:
\[ 2 = \frac{3 + b + 1}{3} \]
Multiplying both sides by 3:
\[ 6 = 3 + b + 1 \]
Simplifying this gives:
\[ 6 = 4 + b \implies b = 6 - 4 = 2 \]

### Step 3: Use the centroid formula for the z-coordinate
The z-coordinate of the centroid \( G \) is given by:
\[ z_G = \frac{z_1 + z_2 + z_3}{3} \]
Substituting the coordinates:
\[ -1 = \frac{2 + (-4) + c}{3} \]
Multiplying both sides by 3:
\[ -3 = 2 - 4 + c \]
This simplifies to:
\[ -3 = -2 + c \implies c = -3 + 2 = -1 \]

### Final Values
Thus, the values are:
\[ a = 3, \quad b = 2, \quad c = -1 \]