In geometry, a perpendicular bisector (đường trung trực in Vietnamese) is a line that divides a line segment into two equal parts at a 90-degree angle. Let’s delve into how to find its equation.
Key Concepts
Line Segment and Midpoint
Consider a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$. The midpoint $M$ of this segment is calculated as:
$M bigg(frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2}bigg)$
Slope of the Line Segment
The slope $m$ of the line segment $AB$ is given by:
$m = frac{y_2 – y_1}{x_2 – x_1}$
Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of $AB$. If the slope of $AB$ is $m$, then the slope of the perpendicular bisector is $-frac{1}{m}$
Deriving the Equation
To find the equation of the perpendicular bisector, we use the point-slope form of a line equation:
$y – y_1 = m(x – x_1)$
Substituting the midpoint $M$ and the slope $-frac{1}{m}$, we get:
$y – frac{y_1 + y_2}{2} = -frac{1}{frac{y_2 – y_1}{x_2 – x_1}} bigg(x – frac{x_1 + x_2}{2}bigg)$
Simplifying, we obtain:
$y – frac{y_1 + y_2}{2} = -frac{x_2 – x_1}{y_2 – y_1} bigg(x – frac{x_1 + x_2}{2}bigg)$
Example
Let’s consider an example where the endpoints of the line segment are $A(2, 3)$ and $B(4, 7)$
Midpoint Calculation:
$M bigg(frac{2 + 4}{2}, frac{3 + 7}{2}bigg) = (3, 5)$Slope of $AB$:
$m = frac{7 – 3}{4 – 2} = 2$Slope of Perpendicular Bisector:
$-frac{1}{2}$Equation of Perpendicular Bisector:
Using the point-slope form:
$y – 5 = -frac{1}{2}(x – 3)$
Simplifying:
$y – 5 = -frac{1}{2}x + frac{3}{2}$
$y = -frac{1}{2}x + frac{13}{2}$
Conclusion
Understanding the equation of the perpendicular bisector involves calculating the midpoint and the slope of the original line segment. This knowledge is crucial for various applications in geometry, such as constructing perpendicular bisectors in triangle circumcircles.