Equation of the Perpendicular Bisector: A Step-by-Step Guide
The equation of the perpendicular bisector of a segment joining two points can be derived through a systematic approach involving several key calculations. This guide will walk you through the process using the points A(2, 2) and B(6, 8).
1. Finding the Midpoint of the Segment
The first step in finding the equation of the perpendicular bisector is to determine the midpoint of the segment joining points A and B. The formula for the midpoint of a segment defined by points ((x_1, y_1)) and ((x_2, y_2)) is given by:
M left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right)
For the points A(2, 2) and B(6, 8), the midpoint is calculated as follows:
M left(frac{2 6}{2}, frac{2 8}{2}right) left(frac{8}{2}, frac{10}{2}right) 4, 5
The midpoint M is therefore (4, 5).
2. Calculating the Slope of Line AB
The next step is to calculate the slope of the line passing through points A and B. The slope (m) of a line through points ((x_1, y_1)) and ((x_2, y_2)) is given by:
m frac{y_2 - y_1}{x_2 - x_1}
For the points A(2, 2) and B(6, 8), the slope is calculated as:
m_{AB} frac{8 - 2}{6 - 2} frac{6}{4} frac{3}{2}
The slope of line AB is (frac{3}{2}).
3. Determining the Slope of the Perpendicular Bisector
To find the slope of the perpendicular bisector, we need to use the negative reciprocal of the slope of line AB. The slope of the perpendicular bisector (m_{text{perpendicular}}) is calculated as:
m_{text{perpendicular}} -frac{1}{m_{AB}} -frac{1}{frac{3}{2}} -frac{2}{3}
The slope of the perpendicular bisector is therefore (-frac{2}{3}).
4. Using the Point-Slope Form to Find the Equation
With the midpoint (4, 5) and the slope (-frac{2}{3}), we can use the point-slope form of a line to find the equation of the perpendicular bisector. The point-slope form of a line is given by:
y - y_1 m(x - x_1)
Plugging in the values, we have:
y - 5 -frac{2}{3}(x - 4)
Now, we simplify this equation:
y - 5 -frac{2}{3}x frac{8}{3}
Multiplying through by 3 to eliminate the fraction, we get:
3y - 15 -2x 8
Finally, rearranging terms to get the equation in the standard form:
3y -2x 23
y -frac{2}{3}x frac{23}{3}
The equation of the perpendicular bisector is thus (y -frac{2}{3}x frac{23}{3}).
Final Summary
The steps to find the equation of the perpendicular bisector of the segment joining points A(2, 2) and B(6, 8) are summarized as follows:
Calculate the midpoint: ((4, 5)) Calculate the slope of the line AB: (frac{3}{2}) Determine the slope of the perpendicular bisector: (-frac{2}{3}) Use the point-slope form to find the equation: (y -frac{2}{3}x frac{23}{3})This method can be applied to any pair of points to find the equation of the perpendicular bisector.