Final answer:
The perpendicular bisector of the line segment joining A(1, 2) and B(7, -4) is the line with the equation y = x - 5, which passes through the midpoint (4, -1) and has a slope of 1.
Step-by-step explanation:
To find the equation of the perpendicular bisector of the line segment joining points A(1, 2) and B(7, -4), you should follow these steps:
- Calculate the midpoint of the line segment AB, which will be a point (xm, ym) on the perpendicular bisector.
- Determine the slope of the line segment AB and then find the negative reciprocal of this slope, which will be the slope of the perpendicular bisector.
- Use the point-slope form of a line equation, with the midpoint and the slope found in steps 1 and 2, to write the equation of the perpendicular bisector.
The midpoint can be found using the formula:
(xm, ym) = ((x1 + x2)/2, (y1 + y2)/2)
In this case, the midpoint is:
(xm, ym) = ((1 + 7)/2, (2 - 4)/2) = (4, -1)
The slope of AB is:
mAB = (y2 - y1) / (x2 - x1) = (-4 - 2)/(7 - 1) = -6/6 = -1
The slope of the perpendicular bisector (mpb) is the negative reciprocal of mAB:
mpb = -1/(-1) = 1
Finally, the equation of the perpendicular bisector using the point-slope form (y - y1) = m(x - x1) and the midpoint (4, -1) is:
y + 1 = 1(x - 4)
Simplifying this, we get the equation:
y = x - 5