The point equidistant from the `O(0,0,0),A(a,0,0),B(0,b,0)` and `C(0,0,c)` has the coordinates

Options

A. `(a,b,c)`
B. `(a//2,b//2,c//2)`
C. `(a//3,b//3,c//3)`
D. `(a//4,b//4,c//4)`

Explanation

To find the point \( P(x, y, z) \) that is equidistant from the points \( O(0, 0, 0) \), \( A(a, 0, 0) \), \( B(0, b, 0) \), and \( C(0, 0, c) \), we need to set up equations based on the distances from point \( P \) to each of these points.

### Step-by-Step Solution:

1. **Distance from \( P \) to \( O \)**:
The distance \( OP \) can be calculated using the distance formula:
\[ OP = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{x^2 + y^2 + z^2} \]

2. **Distance from \( P \) to \( A \)**:
The distance \( AP \) is given by:
\[ AP = \sqrt{(x - a)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{(x - a)^2 + y^2 + z^2} \]

3. **Distance from \( P \) to \( B \)**:
The distance \( BP \) is:
\[ BP = \sqrt{(x - 0)^2 + (y - b)^2 + (z - 0)^2} = \sqrt{x^2 + (y - b)^2 + z^2} \]

4. **Distance from \( P \) to \( C \)**:
The distance \( CP \) is:
\[ CP = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - c)^2} = \sqrt{x^2 + y^2 + (z - c)^2} \]

5. **Setting the distances equal**:
Since \( P \) is equidistant from \( O \), \( A \), \( B \), and \( C \), we can set up the following equations:
\[ OP = AP \implies \sqrt{x^2 + y^2 + z^2} = \sqrt{(x - a)^2 + y^2 + z^2} \]
Squaring both sides:
\[ x^2 + y^2 + z^2 = (x - a)^2 + y^2 + z^2 \]
Simplifying gives:
\[ x^2 = (x - a)^2 \]
Expanding:
\[ x^2 = x^2 - 2ax + a^2 \implies 2ax = a^2 \implies x = \frac{a}{2} \]

6. **Repeating for \( B \)**:
Set \( OP = BP \):
\[ \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (y - b)^2 + z^2} \]
Squaring both sides:
\[ x^2 + y^2 + z^2 = x^2 + (y - b)^2 + z^2 \]
This simplifies to:
\[ y^2 = (y - b)^2 \]
Expanding gives:
\[ y^2 = y^2 - 2by + b^2 \implies 2by = b^2 \implies y = \frac{b}{2} \]

7. **Repeating for \( C \)**:
Set \( OP = CP \):
\[ \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + y^2 + (z - c)^2} \]
Squaring both sides:
\[ x^2 + y^2 + z^2 = x^2 + y^2 + (z - c)^2 \]
This simplifies to:
\[ z^2 = (z - c)^2 \]
Expanding gives:
\[ z^2 = z^2 - 2cz + c^2 \implies 2cz = c^2 \implies z = \frac{c}{2} \]

8. **Final coordinates of point \( P \)**:
Therefore, the coordinates of the point \( P \) that is equidistant from \( O \), \( A \), \( B \), and \( C \) are:
\[ P\left(\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right) \]