Perpendicular Bisector Calculator

What is a Perpendicular Bisector?

A perpendicular bisector is a line that passes through the midpoint of a line segment at a right angle (90 degrees). It divides the original line segment into two equal parts and is perpendicular to it.

How to Calculate the Perpendicular Bisector

To find the perpendicular bisector of a line segment, we follow these steps:

1. Find the midpoint of the line segment
2. Calculate the slope of the original line
3. Determine the negative reciprocal of the slope (perpendicular slope)
4. Use the midpoint and perpendicular slope to form the equation

Formula

The equation of the perpendicular bisector is:

\[ y - y_m = m_p(x - x_m) \]

Where:

• \((x_m, y_m)\) is the midpoint of the original line segment
• \(m_p\) is the slope of the perpendicular bisector

Calculation Steps

1. Find the midpoint: \(x_m = \frac{x_1 + x_2}{2}, y_m = \frac{y_1 + y_2}{2}\)
2. Calculate the slope of the original line: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
3. Find the perpendicular slope: \(m_p = -\frac{1}{m}\)
4. Substitute into the equation: \(y - y_m = m_p(x - x_m)\)

Example and Visual Representation

Let's find the perpendicular bisector of the line segment with endpoints A(1, 1) and B(5, 5):

Step 1: Midpoint = \((\frac{1+5}{2}, \frac{1+5}{2}) = (3, 3)\)

Step 2: Slope of AB = \(\frac{5-1}{5-1} = 1\)

Step 3: Perpendicular slope = \(-\frac{1}{1} = -1\)

Step 4: Equation: \(y - 3 = -1(x - 3)\) or \(y = -x + 6\)

In this diagram, the blue line represents the original line segment AB. The red line is the perpendicular bisector, passing through the midpoint M (shown in green) and forming right angles with AB. Notice how it perfectly balances the original line segment, demonstrating the 'bisector' property.