In the realm of coordinate geometry, perpendicular bisectors play a crucial role in various mathematical applications. A perpendicular bisector is a line that passes through the midpoint of a line segment at a right angle (90 degrees). Understanding how to find the equation of a perpendicular bisector is essential for solving problems involving distances, symmetry, and geometric constructions.
Before we dive into the equations of perpendicular bisectors, it's important to review how to find the midpoint of a line segment. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint formula is:
$$ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) $$
If we have a line segment with endpoints (2, 3) and (6, 7), the midpoint would be:
$$ \left(\frac{2 + 6}{2}, \frac{3 + 7}{2}\right) = (4, 5) $$
To find the equation of a perpendicular bisector, we need to understand the relationship between perpendicular lines. If two lines are perpendicular, their slopes are negative reciprocals of each other. This means that if the slope of one line is $m$, the slope of the line perpendicular to it is $-\frac{1}{m}$.
The product of the slopes of perpendicular lines is always -1.
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