Finding the equation of the perpendicular bisector of a line segment AB involves determining a line that cuts through the midpoint of AB at a 90-degree angle. This line is unique and equidistant from points A and B.
To find the equation of the perpendicular bisector of a line segment AB, follow these sequential steps:
What is a Perpendicular Bisector?
A perpendicular bisector is a line that intersects another line segment at its midpoint and forms a right angle (90 degrees) with it. In the context of a line segment AB, every point on its perpendicular bisector is equidistant from point A and point B.
Steps to Find the Equation of the Perpendicular Bisector
The process involves calculating the midpoint of the segment, finding its slope, determining the slope of the perpendicular line, and finally using the point-gradient formula.
1. Find the Midpoint of AB
The midpoint of a line segment is the average of the coordinates of its endpoints. If the coordinates of A are $(x_1, y_1)$ and B are $(x_2, y_2)$, the midpoint M is given by:
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
This midpoint will be a point on the perpendicular bisector. For more details on calculating midpoints, you can refer to resources on the midpoint formula.
2. Find the Gradient (Slope) of AB
The gradient (or slope) of the line segment AB tells us its steepness. It is calculated as the change in y-coordinates divided by the change in x-coordinates.
$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}$
Understanding how to calculate slopes is fundamental in coordinate geometry. You can find more information on the slope of a line.
3. Find the Perpendicular Gradient
Two lines are perpendicular if the product of their gradients is -1. If the gradient of AB is $m{AB}$, then the gradient of the perpendicular bisector, $m{\perp}$, will be its negative reciprocal:
$m{\perp} = -\frac{1}{m{AB}}$
- Special Cases:
- If $m_{AB}$ is 0 (a horizontal line), the perpendicular bisector will be a vertical line with an undefined slope (equation $x = \text{constant}$).
- If $m_{AB}$ is undefined (a vertical line), the perpendicular bisector will be a horizontal line with a slope of 0 (equation $y = \text{constant}$).
For a deeper dive into perpendicular lines and their slopes, explore resources on perpendicular lines.
4. Use the Point-Gradient Formula to Find the Equation
Now that you have a point on the perpendicular bisector (the midpoint M) and its gradient ($m_{\perp}$), you can use the point-gradient (or point-slope) formula to write the equation of the line:
$y - yM = m{\perp} (x - x_M)$
Where $(x_M, yM)$ are the coordinates of the midpoint and $m{\perp}$ is the perpendicular gradient. This formula is a powerful tool for finding the equation of a line. Learn more about the point-slope form.
You can then rearrange this equation into other forms, such as the standard form ($Ax + By + C = 0$) or the slope-intercept form ($y = mx + c$), if required.
Example: Finding the Perpendicular Bisector of AB
Let's find the equation of the perpendicular bisector for the line segment AB with endpoints A(1, 2) and B(5, 6).
-
Find the Midpoint (M) of AB:
$x_M = \frac{1 + 5}{2} = \frac{6}{2} = 3$
$y_M = \frac{2 + 6}{2} = \frac{8}{2} = 4$
So, the midpoint M is (3, 4). -
Find the Gradient ($m_{AB}$) of AB:
$m_{AB} = \frac{6 - 2}{5 - 1} = \frac{4}{4} = 1$ -
Find the Perpendicular Gradient ($m_{\perp}$):
$m{\perp} = -\frac{1}{m{AB}} = -\frac{1}{1} = -1$ -
Use the Point-Gradient Formula:
Using M(3, 4) and $m_{\perp} = -1$:
$y - yM = m{\perp} (x - x_M)$
$y - 4 = -1 (x - 3)$
$y - 4 = -x + 3$
$y = -x + 3 + 4$
$y = -x + 7$The equation of the perpendicular bisector of AB is $y = -x + 7$.
Practical Insights and Applications
- Geometric Construction: The perpendicular bisector is key in geometric constructions, such as finding the center of a circle that passes through three given points (circumcenter of a triangle).
- Optimization Problems: It can be used in problems where you need to find a location equidistant from two points.
- Coordinate Geometry Foundation: Mastering this concept reinforces understanding of slopes, midpoints, and linear equations, which are fundamental in higher-level mathematics.
By systematically following these steps, you can accurately determine the equation of any perpendicular bisector for a given line segment.
