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, we need to find the coordinates of point P (a, b, c) given that the centroid of tetrahedron OABC is (1, 2, -2). The vertices A, B, and C are given as follows:

- A = (a, 2, 3)
- B = (1, b, 2)
- C = (2, 1, c)

The centroid G of tetrahedron OABC is calculated using the formula:

\[ G = \left( \frac{x_1 + x_2 + x_3 + x_4}{4}, \frac{y_1 + y_2 + y_3 + y_4}{4}, \frac{z_1 + z_2 + z_3 + z_4}{4} \right) \]

Where O is the origin (0, 0, 0). Thus, the coordinates of the centroid can be expressed as:

\[ G = \left( \frac{a + 1 + 2 + 0}{4}, \frac{2 + b + 1 + 0}{4}, \frac{3 + 2 + c + 0}{4} \right) \]

Setting this equal to the given centroid (1, 2, -2), we have:

1. For the x-coordinate:
\[ \frac{a + 1 + 2 + 0}{4} = 1 \]
Multiplying both sides by 4:
\[ a + 3 = 4 \implies a = 1 \]

2. For the y-coordinate:
\[ \frac{2 + b + 1 + 0}{4} = 2 \]
Multiplying both sides by 4:
\[ 2 + b + 1 = 8 \implies b + 3 = 8 \implies b = 5 \]

3. For the z-coordinate:
\[ \frac{3 + 2 + c + 0}{4} = -2 \]
Multiplying both sides by 4:
\[ 3 + 2 + c = -8 \implies c + 5 = -8 \implies c = -13 \]

Now we have the coordinates of point P:
\[ P = (1, 5, -13) \]

Next, we need to find the distance of point P from the origin O (0, 0, 0). The distance formula between two points in 3D space is given by:

\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Substituting the coordinates of P and O into the formula:
\[ D = \sqrt{(1 - 0)^2 + (5 - 0)^2 + (-13 - 0)^2} \]
\[ D = \sqrt{1^2 + 5^2 + (-13)^2} \]
\[ D = \sqrt{1 + 25 + 169} \]
\[ D = \sqrt{195} \]

Thus, the distance of point P from the origin is \(\sqrt{195}\).

### Summary of Steps:
1. Write the formula for the centroid of tetrahedron OABC.
2. Set up equations for the x, y, and z coordinates of the centroid.
3. Solve for a, b, and c using the equations.
4. Substitute the values of a, b, and c into point P.
5. Use the distance formula to find the distance from point P to the origin.