What is the equation for a perpendicular bisector?
Understand the Problem
The question is asking for the equation that represents a perpendicular bisector, which is a line that divides a segment into two equal parts at a right angle. To find this equation, we will identify the midpoint of the segment and then determine the slope of the line perpendicular to the segment.
Answer
y - \frac{y_1 + y_2}{2} = -\frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} \left( x - \frac{x_1 + x_2}{2} \right)
Answer for screen readers
With endpoints $(x_1, y_1)$ and $(x_2, y_2)$, the equation of the perpendicular bisector is: $$ y - \frac{y_1 + y_2}{2} = -\frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} \left( x - \frac{x_1 + x_2}{2} \right) $$
Steps to Solve
- Identify the coordinates of the endpoints of the segment
You will need the coordinates of the two endpoints, which we will call $(x_1, y_1)$ and $(x_2, y_2)$.
- Calculate the midpoint of the segment
The midpoint $M$ can be found using the midpoint formula:
$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$
- Determine the slope of the segment
The slope $m$ of the segment connecting the endpoints can be found using the slope formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
- Calculate the slope of the perpendicular bisector
The slope $m_\perp$ of the line that is perpendicular to the original segment is the negative reciprocal of $m$.
$$m_\perp = -\frac{1}{m}$$
- Use the point-slope form of the equation of a line
With the slope of the perpendicular bisector and the coordinates of the midpoint, use the point-slope form to write the equation of the perpendicular bisector:
$$y - y_\text{mid} = m_\perp (x - x_\text{mid})$$
This can be simplified to get the final equation of the perpendicular bisector.
With endpoints $(x_1, y_1)$ and $(x_2, y_2)$, the equation of the perpendicular bisector is: $$ y - \frac{y_1 + y_2}{2} = -\frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} \left( x - \frac{x_1 + x_2}{2} \right) $$
More Information
The perpendicular bisector not only divides the segment into two equal parts but also forms a right angle with the segment.
Tips
A common mistake is forgetting to take the negative reciprocal when finding the slope of the perpendicular bisector.
