Question

Find the co-ordinates of the point divided by x-axis to the line segment joining the points \( (1, -5) \) and \( (-4, 5) \).

Hint:

When a point divides a line segment, use the section formula to calculate the coordinates. For a point on the x-axis, the y-coordinate will always be 0.

Answer 1

The x-axis divides the line segment joining two points in such a way that the y-coordinate of the point is 0. Let the required point be \( P(x, 0) \). We can use the section formula to find the coordinates of this point. The section formula for a point dividing a line segment in the ratio \( m:n \) is given by: \[ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n}. \] In this case, the point \( P \) divides the segment joining \( (1, -5) \) and \( (-4, 5) \) in the ratio \( m:n = 1:1 \) (since the point lies on the x-axis). Therefore, the coordinates of point \( P \) are: \[ x = \frac{1 \times (-4) + 1 \times 1}{1 + 1} = \frac{-4 + 1}{2} = \frac{-3}{2} = -1.5. \] \[ y = \frac{1 \times 5 + 1 \times (-5)}{1 + 1} = \frac{5 - 5}{2} = 0. \] Thus, the coordinates of the point are \( (-1.5, 0) \).

Conclusion: The coordinates of the point are \( (-1.5, 0) \).