Find the coordinates of the point which divies the line segment joining the points (1, 3, 5) and
(-2, 1, 0) in the ratio 1 : 2 internally.

Answer

To find the coordinates of the point that divides the line segment joining the points \( A(1, 3, 5) \) and \( B(-2, 1, 0) \) in the ratio \( 1:2 \) internally, we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points and the Ratio**:
- Let the coordinates of point \( A \) be \( (x_1, y_1, z_1) = (1, 3, 5) \).
- Let the coordinates of point \( B \) be \( (x_2, y_2, z_2) = (-2, 1, 0) \).
- The ratio in which the point divides the line segment is \( m:n = 1:2 \).

2. **Apply the Section Formula**:
The section formula for a point \( R(x, y, z) \) that divides the line segment joining points \( A \) and \( B \) in the ratio \( m:n \) is given by:
\[ R\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

3. **Substitute the Values**:
- For \( x \)-coordinate:
\[ x = \frac{1 \cdot (-2) + 2 \cdot 1}{1 + 2} = \frac{-2 + 2}{3} = \frac{0}{3} = 0 \]
- For \( y \)-coordinate:
\[ y = \frac{1 \cdot 1 + 2 \cdot 3}{1 + 2} = \frac{1 + 6}{3} = \frac{7}{3} \]
- For \( z \)-coordinate:
\[ z = \frac{1 \cdot 0 + 2 \cdot 5}{1 + 2} = \frac{0 + 10}{3} = \frac{10}{3} \]

4. **Combine the Coordinates**:
Thus, the coordinates of point \( R \) that divides the line segment in the ratio \( 1:2 \) are:
\[ R\left(0, \frac{7}{3}, \frac{10}{3}\right) \]

### Final Answer:
The coordinates of the point that divides the line segment joining \( (1, 3, 5) \) and \( (-2, 1, 0) \) in the ratio \( 1:2 \) internally are \( \left(0, \frac{7}{3}, \frac{10}{3}\right) \).