Assertion `(A):` If `(-1,3,2) and (5,3,2)` are respectively the orthocentre and circumcentre of a triangle, then `(3,3,2)` is its centroid. Reason `(R):` Centroid of a triangle divides the line segment joining the orthocentre and the circumcentre in the ratio `1:2`,

Options

A. A and R are true and R is correct explanation to A.
B. A and R are true but R is not the correct explanation to A.
C. A is true, R is false.
D. A is false. R is true.

Answer

To solve the problem, we will verify the assertion and reason provided in the question step by step.

### Step 1: Identify the Points
We are given two points:
- Orthocenter \( H = (-1, 3, 2) \)
- Circumcenter \( O = (5, 3, 2) \)

### Step 2: Use the Section Formula
The centroid \( G \) of a triangle divides the line segment joining the orthocenter and circumcenter in the ratio \( 2:1 \). According to the section formula, if a point \( G \) divides the line segment joining points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of point \( G \) can be calculated as follows:

\[ G_x = \frac{mx_2 + nx_1}{m+n}, \quad G_y = \frac{my_2 + ny_1}{m+n}, \quad G_z = \frac{mz_2 + nz_1}{m+n} \]

### Step 3: Assign Values
Here, we have:
- \( A = H = (-1, 3, 2) \)
- \( B = O = (5, 3, 2) \)
- \( m = 2 \) (for circumcenter)
- \( n = 1 \) (for orthocenter)

### Step 4: Calculate the Coordinates of the Centroid
Using the section formula, we can calculate the coordinates of \( G \):

1. **Calculate \( G_x \)**:
\[ G_x = \frac{2 \cdot 5 + 1 \cdot (-1)}{2 + 1} = \frac{10 - 1}{3} = \frac{9}{3} = 3 \]

2. **Calculate \( G_y \)**:
\[ G_y = \frac{2 \cdot 3 + 1 \cdot 3}{2 + 1} = \frac{6 + 3}{3} = \frac{9}{3} = 3 \]

3. **Calculate \( G_z \)**:
\[ G_z = \frac{2 \cdot 2 + 1 \cdot 2}{2 + 1} = \frac{4 + 2}{3} = \frac{6}{3} = 2 \]

### Step 5: Result
Thus, the coordinates of the centroid \( G \) are:
\[ G = (3, 3, 2) \]

### Step 6: Conclusion
The assertion states that if \( H \) and \( O \) are the orthocenter and circumcenter respectively, then \( G = (3, 3, 2) \) is the centroid, which we have verified to be true.

The reason states that the centroid divides the line segment joining the orthocenter and circumcenter in the ratio \( 1:2 \), which is incorrect. The correct ratio is \( 2:1 \).

### Final Answer
- Assertion \( (A) \) is **True**.
- Reason \( (R) \) is **False**.