Perpendicular bisector center
circumcenter
The perpendicular bisector of a line segment is a line that cuts the segment into two equal halves. The center of the perpendicular bisector is the point where the line intersects the midpoint of the line segment.
To find the center of the perpendicular bisector of a line segment with endpoints A(x1, y1) and B(x2, y2), we need to follow these steps:
1. Find the midpoint of the line segment using the formula:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
2. Calculate the slope of the line segment using the formula:
Slope = (y2 – y1)/(x2 – x1)
3. Find the negative reciprocal of the slope. This is the slope of the perpendicular bisector.
4. Use the point-slope form of the equation of a line to find the equation of the perpendicular bisector.
Equation of the perpendicular bisector: y – ym = mperp(x – xm)
where mperp = -1/slope and (xm, ym) is the midpoint.
5. Simplify the equation to slope-intercept form (y = mx + b) to find the y-intercept, which is the center of the perpendicular bisector.
Example:
Find the center of the perpendicular bisector of the line segment with endpoints A(2, 3) and B(6, 7).
1. Midpoint = [(2 + 6)/2, (3 + 7)/2] = (4, 5)
2. Slope = (7 – 3)/(6 – 2) = 1
3. Negative reciprocal of the slope: -1/1 = -1
4. Equation of the perpendicular bisector: y – 5 = -1(x – 4) or y = -x + 9
5. Slope-intercept form: y = -1x + 9. The y-intercept is (0, 9), which is the center of the perpendicular bisector.
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