If the distance between centroid and orthocentre of a triangle is 12 cm then find the distance between its orthocentre and circumcentre.

Answer

To solve the problem, we will use the relationship between the centroid (G), orthocenter (H), and circumcenter (O) of a triangle.

### Step-by-Step Solution:

1. **Understand the Relationship**: In any triangle, the distances between the centroid (G), orthocenter (H), and circumcenter (O) are related by the formula:
\[ GH = \frac{2}{3} OH \]
where \( GH \) is the distance between the centroid and orthocenter, and \( OH \) is the distance between the orthocenter and circumcenter.

2. **Given Information**: We know that the distance between the centroid and orthocenter \( GH \) is given as 12 cm. Therefore:
\[ GH = 12 \text{ cm} \]

3. **Set Up the Equation**: According to the relationship, we can express \( OH \) in terms of \( GH \):
\[ GH = \frac{2}{3} OH \]
Rearranging gives:
\[ OH = \frac{3}{2} GH \]

4. **Substitute the Known Value**: Now, substitute \( GH = 12 \text{ cm} \) into the equation:
\[ OH = \frac{3}{2} \times 12 \]

5. **Calculate \( OH \)**:
\[ OH = \frac{3 \times 12}{2} = \frac{36}{2} = 18 \text{ cm} \]

6. **Final Answer**: Therefore, the distance between the orthocenter and circumcenter \( OH \) is:
\[ OH = 18 \text{ cm} \]