Find the co-ordinates of a point which divides the line joining the points (5, 3) and (10, 8) in the ratio 2 : 3 internally.

Answer

To find the coordinates of a point that divides the line segment joining the points (5, 3) and (10, 8) in the ratio 2:3 internally, we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points and the Ratio:**
- Let the points be \( A(5, 3) \) and \( B(10, 8) \).
- The ratio in which the point divides the line segment is \( m:n = 2:3 \).

2. **Assign Values:**
- Here, \( x_1 = 5 \), \( y_1 = 3 \), \( x_2 = 10 \), \( y_2 = 8 \), \( m = 2 \), and \( n = 3 \).

3. **Use the Section Formula:**
- The coordinates \( (x, y) \) of the point that divides the line segment in the ratio \( m:n \) are given by:
\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \]
\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \]

4. **Calculate the x-coordinate:**
- Substitute the values into the formula:
\[ x = \frac{2 \cdot 10 + 3 \cdot 5}{2 + 3} = \frac{20 + 15}{5} = \frac{35}{5} = 7 \]

5. **Calculate the y-coordinate:**
- Substitute the values into the formula:
\[ y = \frac{2 \cdot 8 + 3 \cdot 3}{2 + 3} = \frac{16 + 9}{5} = \frac{25}{5} = 5 \]

6. **Final Coordinates:**
- Therefore, the coordinates of the point that divides the line segment in the ratio 2:3 are \( (7, 5) \).

### Final Answer:
The coordinates of the required point are \( (7, 5) \).

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