How to find the perpendicular bisector of a line?
Understand the Problem
The question is asking for the method to determine the perpendicular bisector of a line segment. This typically involves finding the midpoint of the segment and determining the slope of the line that is perpendicular to it.
Answer
The equation of the perpendicular bisector is: $$ y - \frac{y_1 + y_2}{2} = -\frac{1}{\left( \frac{y_2 - y_1}{x_2 - x_1} \right)}\left(x - \frac{x_1 + x_2}{2}\right) $$
Answer for screen readers
The equation of the perpendicular bisector of the line segment between points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by: $$ y - \frac{y_1 + y_2}{2} = -\frac{1}{\left( \frac{y_2 - y_1}{x_2 - x_1} \right)}\left(x - \frac{x_1 + x_2}{2}\right) $$
Steps to Solve
- Identify the endpoints of the line segment
Let's say the endpoints of the line segment are given as points $A(x_1, y_1)$ and $B(x_2, y_2)$.
- Find the midpoint of the segment
The midpoint $M$ of the line segment can be calculated using the formula: $$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$
- Calculate the slope of the line segment
The slope $m$ of the line segment can be determined using the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
- Determine the slope of the perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment. If the slope of the line segment is $m$, then: $$ m_{\perp} = -\frac{1}{m} $$
- Write the equation of the perpendicular bisector
Using the point-slope form of the equation of a line, the equation for the perpendicular bisector can be expressed as: $$ y - y_M = m_{\perp}(x - x_M) $$
Here, $(x_M, y_M)$ are the coordinates of the midpoint found earlier.
The equation of the perpendicular bisector of the line segment between points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by: $$ y - \frac{y_1 + y_2}{2} = -\frac{1}{\left( \frac{y_2 - y_1}{x_2 - x_1} \right)}\left(x - \frac{x_1 + x_2}{2}\right) $$
More Information
The perpendicular bisector divides the line segment into two equal parts at a right angle. It's an important concept in geometry, especially in the study of triangles and circles.
Tips
- Confusing the slopes: A common mistake is to incorrectly calculate the slope of the original line segment, which leads to an incorrect slope for the perpendicular bisector. To avoid this, double-check the calculations and remember that the slope of the perpendicular bisector is the negative reciprocal.