If (1, 1, 1) is the centroid of the triangle with vertices (2a, 3, 7), (-4, 3b, -8) and (8, 4, 6c),
find a, b, c.

Answer

To find the values of \( a \), \( b \), and \( c \) given that the point \( (1, 1, 1) \) is the centroid of the triangle with vertices \( (2a, 3, 7) \), \( (-4, 3b, -8) \), and \( (8, 4, 6c) \), we can use the formula for the centroid of a triangle in three-dimensional space.

### Step 1: Write the formula for the centroid
The centroid \( G \) of a triangle with vertices \( (x_1, y_1, z_1) \), \( (x_2, y_2, z_2) \), and \( (x_3, y_3, z_3) \) 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: Set up the equations for the coordinates
Given the vertices:
- \( (2a, 3, 7) \)
- \( (-4, 3b, -8) \)
- \( (8, 4, 6c) \)

We can set up the equations for the centroid coordinates:
1. For the x-coordinate:
\[ \frac{2a - 4 + 8}{3} = 1 \]
2. For the y-coordinate:
\[ \frac{3 + 3b + 4}{3} = 1 \]
3. For the z-coordinate:
\[ \frac{7 - 8 + 6c}{3} = 1 \]

### Step 3: Solve for \( a \)
From the x-coordinate equation:
\[ \frac{2a + 4}{3} = 1 \]
Multiplying both sides by 3:
\[ 2a + 4 = 3 \]
Subtracting 4 from both sides:
\[ 2a = -1 \]
Dividing by 2:
\[ a = -\frac{1}{2} \]

### Step 4: Solve for \( b \)
From the y-coordinate equation:
\[ \frac{3 + 3b + 4}{3} = 1 \]
Multiplying both sides by 3:
\[ 3 + 3b + 4 = 3 \]
Combining like terms:
\[ 3b + 7 = 3 \]
Subtracting 7 from both sides:
\[ 3b = -4 \]
Dividing by 3:
\[ b = -\frac{4}{3} \]

### Step 5: Solve for \( c \)
From the z-coordinate equation:
\[ \frac{7 - 8 + 6c}{3} = 1 \]
Multiplying both sides by 3:
\[ 7 - 8 + 6c = 3 \]
Simplifying:
\[ -1 + 6c = 3 \]
Adding 1 to both sides:
\[ 6c = 4 \]
Dividing by 6:
\[ c = \frac{2}{3} \]

### Final Values
Thus, the values are:
\[ a = -\frac{1}{2}, \quad b = -\frac{4}{3}, \quad c = \frac{2}{3} \]