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Perpendicular Bisector Equation:
Where \( (x_m, y_m) \) is the midpoint and \( m_p \) is the perpendicular slope
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A perpendicular bisector is a line that cuts another line segment into two equal parts at a 90-degree angle. In coordinate geometry, it's the line that passes through the midpoint of a segment and is perpendicular to that segment.
The calculator uses the following mathematical principles:
Midpoint: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Slope of original segment: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Perpendicular slope: \( m_p = -\frac{1}{m} \)
Equation: \( y - y_m = m_p(x - x_m) \)
Where:
Details: Perpendicular bisectors are fundamental in geometry, used in constructing circumcircles, finding equidistant points, solving geometric problems, and in various engineering and architectural applications.
Tips: Enter the coordinates of two distinct points. The calculator will find the equation of the perpendicular bisector. For vertical or horizontal lines, special cases are handled appropriately.
Q1: What if the two points are the same?
A: The calculator requires distinct points to form a valid line segment. Identical points will result in an error.
Q2: How is the perpendicular slope calculated?
A: The perpendicular slope is the negative reciprocal of the original slope. If the original slope is m, the perpendicular slope is -1/m.
Q3: What happens with vertical lines?
A: For vertical lines (x1 = x2), the perpendicular bisector will be a horizontal line (y = constant).
Q4: What happens with horizontal lines?
A: For horizontal lines (y1 = y2), the perpendicular bisector will be a vertical line (x = constant).
Q5: Can I use this for 3D coordinates?
A: This calculator is designed for 2D coordinate geometry only. For 3D, you would need to find the perpendicular bisecting plane.