If distance betweenn orthocentre and circumcentre of a triangle is 6 cm then what is the distance between its centroid and circumcentre ?

Options

A. 4 cm
B. 2 cm
C. `8/3` cm
D. `4/3` cm

Answer

To solve the problem, we need to understand the relationship between the Orthocenter (H), Circumcenter (O), and Centroid (G) of a triangle. These points are collinear and divide the line segment OH in a specific ratio.

### Step-by-Step Solution:

1. **Understanding the Points**:
- The Orthocenter (H), Circumcenter (O), and Centroid (G) of a triangle are collinear.
- The ratio in which G divides the line segment OH is 1:2.

2. **Given Information**:
- The distance between the Orthocenter (H) and Circumcenter (O) is given as 6 cm. This means the length of segment OH = 6 cm.

3. **Setting Up the Ratio**:
- Since G divides OH in the ratio of 1:2, we can denote the distances as follows:
- OG = 1 part
- GH = 2 parts
- Therefore, the total parts = 1 + 2 = 3 parts.

4. **Finding the Length of Each Part**:
- If the total distance OH is 6 cm and it is divided into 3 equal parts, then each part is:
\[ \text{Length of each part} = \frac{6 \text{ cm}}{3} = 2 \text{ cm} \]

5. **Finding the Distance OG**:
- Since OG is 1 part, we have:
\[ OG = 1 \times 2 \text{ cm} = 2 \text{ cm} \]

6. **Conclusion**:
- The distance between the Centroid (G) and Circumcenter (O) is 2 cm.

### Final Answer:
The distance between the Centroid and Circumcenter is **2 cm**.