The coordinates of the centrid of a triangle having its circumcentre and orthocenrtre at (7/2,5/2) and (2,1) respectively, are

Options

A. (3,2)
B. (13/6,3/2)
C. (5/2,3/2)
D. (3/2,5/2)

Answer

To find the coordinates of the centroid of a triangle given its circumcenter and orthocenter, we can use the section formula. The centroid (G) divides the line segment joining the orthocenter (H) and circumcenter (O) in the ratio 2:1.

### Step-by-Step Solution:

1. **Identify the Coordinates**:
- Circumcenter (O) = \( \left( \frac{7}{2}, \frac{5}{2} \right) \)
- Orthocenter (H) = \( (2, 1) \)

2. **Use the Section Formula**:
The coordinates of the centroid (G) can be calculated using the section formula. If a point G divides the line segment joining points (x1, y1) and (x2, y2) in the ratio m:n, the coordinates of G are given by:
\[ G_x = \frac{mx_2 + nx_1}{m+n} \]
\[ G_y = \frac{my_2 + ny_1}{m+n} \]

Here, we have:
- \( m = 2 \) (for the circumcenter)
- \( n = 1 \) (for the orthocenter)
- \( (x_1, y_1) = (2, 1) \) (orthocenter)
- \( (x_2, y_2) = \left( \frac{7}{2}, \frac{5}{2} \right) \) (circumcenter)

3. **Calculate the x-coordinate of G**:
\[ G_x = \frac{2 \cdot \frac{7}{2} + 1 \cdot 2}{2 + 1} \]
Simplifying:
\[ G_x = \frac{7 + 2}{3} = \frac{9}{3} = 3 \]

4. **Calculate the y-coordinate of G**:
\[ G_y = \frac{2 \cdot \frac{5}{2} + 1 \cdot 1}{2 + 1} \]
Simplifying:
\[ G_y = \frac{5 + 1}{3} = \frac{6}{3} = 2 \]

5. **Final Coordinates of the Centroid**:
Therefore, the coordinates of the centroid (G) are:
\[ G = (3, 2) \]

### Final Answer:
The coordinates of the centroid of the triangle are \( (3, 2) \).