If the centroid of tetrahedron OABC where A,B,C are given by (a,2,3),(1,b,2) and (2,1,c) respectively is (1,2,−2), then distance of P(a,b,c) from origin is

Answer

To solve the problem step by step, we need to find the coordinates of points A, B, and C based on the given centroid of the tetrahedron OABC and then calculate the distance of point P(a, b, c) from the origin.

### Step 1: Identify the coordinates of the points
- The coordinates of point O (origin) are (0, 0, 0).
- The coordinates of point A are given as (a, 2, 3).
- The coordinates of point B are given as (1, b, 2).
- The coordinates of point C are given as (2, 1, c).
- The centroid of the tetrahedron OABC is given as (1, 2, -2).

### Step 2: Use the centroid formula
The formula for the centroid \( G \) of a tetrahedron with vertices at points \( O, A, B, C \) is given by:
\[ G = \left( \frac{x_O + x_A + x_B + x_C}{4}, \frac{y_O + y_A + y_B + y_C}{4}, \frac{z_O + z_A + z_B + z_C}{4} \right) \]

### Step 3: Set up the equations for each coordinate
1. For the x-coordinate:
\[ \frac{0 + a + 1 + 2}{4} = 1 \]
Simplifying this gives:
\[ \frac{a + 3}{4} = 1 \implies a + 3 = 4 \implies a = 1 \]

2. For the y-coordinate:
\[ \frac{0 + 2 + b + 1}{4} = 2 \]
Simplifying this gives:
\[ \frac{b + 3}{4} = 2 \implies b + 3 = 8 \implies b = 5 \]

3. For the z-coordinate:
\[ \frac{0 + 3 + 2 + c}{4} = -2 \]
Simplifying this gives:
\[ \frac{5 + c}{4} = -2 \implies 5 + c = -8 \implies c = -13 \]

### Step 4: Determine the coordinates of point P
Now we have the coordinates of points A, B, and C:
- \( A = (1, 2, 3) \)
- \( B = (1, 5, 2) \)
- \( C = (2, 1, -13) \)

Thus, point P(a, b, c) is \( (1, 5, -13) \).

### Step 5: Calculate the distance from the origin
The distance \( d \) from the origin (0, 0, 0) to point P(a, b, c) is given by the distance formula:
\[ d = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(1 - 0)^2 + (5 - 0)^2 + (-13 - 0)^2} \]
Calculating this gives:
\[ d = \sqrt{1^2 + 5^2 + (-13)^2} = \sqrt{1 + 25 + 169} = \sqrt{195} \]

### Final Answer
The distance of point P(a, b, c) from the origin is \( \sqrt{195} \).