Answer:
Sure, let's consider the equation of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \). The equation of this line can be expressed as:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope of the line, given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Now, for the perpendicular bisector of this line segment, we need to find the midpoint of the segment and its slope. Let \( (x_m, y_m) \) be the midpoint of the segment, then:
\[ x_m = \frac{x_1 + x_2}{2} \]
\[ y_m = \frac{y_1 + y_2}{2} \]
The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line. So, the slope of the perpendicular bisector, \( m_{\text{perpendicular}} \), is:
\[ m_{\text{perpendicular}} = -\frac{1}{m} \]
Thus, the equation of the perpendicular bisector passing through \( (x_m, y_m) \) with slope \( m_{\text{perpendicular}} \) is:
\[ y - y_m = m_{\text{perpendicular}}