Answer:
To find the equation of the perpendicular bisector of AB with A(-6, 8) and B(-1, 3), we need to first find the midpoint of AB, and then find the slope of the line perpendicular to AB that passes through the midpoint.
Midpoint of AB:
The midpoint of AB is given by the formula:
((x1 + x2)/2, (y1 + y2)/2)
Substituting the values, we get:
((-6 + (-1))/2, (8 + 3)/2) = (-3.5, 5.5)
So, the midpoint of AB is (-3.5, 5.5).
Slope of the line AB:
The slope of AB is given by the formula:
m = (y2 - y1)/(x2 - x1)
Substituting the values, we get:
m = (3 - 8)/(-1 - (-6)) = 5/5 = 1
So, the slope of AB is 1.
Slope of the perpendicular bisector:
Since the perpendicular bisector of AB is perpendicular to AB, its slope will be the negative reciprocal of the slope of AB. Therefore, the slope of the perpendicular bisector is -1.
Equation of the perpendicular bisector:
Now that we have the slope of the perpendicular bisector (-1) and the midpoint of AB (-3.5, 5.5), we can use the point-slope form of a line to find the equation of the perpendicular bisector:
y - y1 = m(x - x1)
Substituting the values, we get:
y - 5.5 = -1(x - (-3.5))
y - 5.5 = -x - 3.5
y = -x + 2
Therefore, the equation of the line that is the perpendicular bisector of AB with A(-6, 8) and B(-1, 3) is y = -x + 2, which is option (D).
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