What are the coordinates of the point where the perpendicular bisector of the line segment between A(2, -4) and B(-6, -2) intersects the x-axis?

To find the coordinates of the point where the perpendicular bisector of the line segment between points A(2, -4) and B(-6, -2) intersects the x-axis, we first need to determine the midpoint of the segment AB and then find the slope of the segment and the slope of the perpendicular bisector.

1. Finding the Midpoint of AB:

The midpoint M of a line segment with endpoints A(x1, y1) and B(x2, y2) is calculated as:

[

M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right)

]

So, substituting the coordinates:

[

M = \left( \frac{2 + (-6)}{2}, \frac{-4 + (-2)}{2} \right) = \left( \frac{-4}{2}, \frac{-6}{2} \right) = (-2, -3)

]

2. Finding the Slope of AB:

The slope of a line is found using the formula:

[

\text{slope} = \frac