Question

Find the coordinates of the point which divides the line segment formed by joining the points \( (-1, 7) \) and \( (4, -3) \) in the ratio \( 2 : 3 \).

Hint:

To find the coordinates of a point dividing a line segment, use the section formula.

Answer 1

The formula to find the coordinates of a point dividing a line segment in a ratio \( m : n \) is given by: \[ \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \] Here, the given points are \( (-1, 7) \) and \( (4, -3) \), and the ratio is \( 2 : 3 \). Let \( (x, y) \) be the coordinates of the point that divides the line segment. \[ x = \frac{2 \times 4 + 3 \times (-1)}{2 + 3} = \frac{8 - 3}{5} = \frac{5}{5} = 1 \] \[ y = \frac{2 \times (-3) + 3 \times 7}{2 + 3} = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \] Thus, the coordinates of the point are \( (1, 3) \).
Conclusion:
The coordinates of the point dividing the line segment in the ratio \( 2 : 3 \) are \( (1, 3) \).