If the orthocentre and centroid of a triangle are (-3, 5, 1) and (3, 3, -1) respectively, then its circumcentre is

Options

A. (6, 2, -2)
B. (1, 4, 0)
C. (6, 2, 2)
D. (6, -2, 2)

Answer

To find the circumcenter of the triangle given its orthocenter and centroid, we can use the property that the centroid divides the line segment joining the orthocenter (H) and circumcenter (O) in the ratio 2:1.

### Step-by-Step Solution:

1. **Identify the Given Points:**
- Orthocenter (H) = (-3, 5, 1)
- Centroid (G) = (3, 3, -1)
- Circumcenter (O) = (x, y, z) (unknown)

2. **Use the Ratio Property:**
The centroid divides the line segment joining the orthocenter and circumcenter in the ratio 2:1. This can be expressed as:
\[ G = \left( \frac{2O + H}{3} \right) \]

3. **Set Up the Equations:**
For the x-coordinate:
\[ 3 = \frac{2x + (-3)}{3} \]
For the y-coordinate:
\[ 3 = \frac{2y + 5}{3} \]
For the z-coordinate:
\[ -1 = \frac{2z + 1}{3} \]

4. **Solve for x:**
Multiply both sides of the x-coordinate equation by 3:
\[ 9 = 2x - 3 \]
Rearranging gives:
\[ 2x = 12 \implies x = 6 \]

5. **Solve for y:**
Multiply both sides of the y-coordinate equation by 3:
\[ 9 = 2y + 5 \]
Rearranging gives:
\[ 2y = 4 \implies y = 2 \]

6. **Solve for z:**
Multiply both sides of the z-coordinate equation by 3:
\[ -3 = 2z + 1 \]
Rearranging gives:
\[ 2z = -4 \implies z = -2 \]

7. **Conclusion:**
The circumcenter O of the triangle is:
\[ O = (6, 2, -2) \]

### Final Answer:
The circumcenter of the triangle is (6, 2, -2).

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