Which equation represents the perpendicular bisector of AB whose endpoints are A(8,2) and B(0,6)? *
the midpoint of AB is ... [(8 + 0)/2 , (2 + 6)/2]
the slope of AB is ... (6 - 2) / (0 - 8)
the perpendicular bisector passes through the midpoint
... and has a slope that is the negative-reciprocal of AB's slope
To find the equation of the perpendicular bisector of segment AB, we need to find two things: the midpoint of the segment and the slope of the perpendicular bisector.
Step 1: Finding the midpoint
To find the midpoint of segment AB, we use the formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Given the endpoints A(8,2) and B(0,6), we can substitute the values into the formula:
Midpoint = ((8 + 0) / 2, (2 + 6) / 2)
Midpoint = (8 / 2, 8 / 2)
Midpoint = (4, 4)
So, the midpoint of segment AB is M(4,4).
Step 2: Finding the slope of the perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of segment AB.
To find the slope, we use the formula:
Slope (m) = (y2 - y1) / (x2 - x1)
Given the endpoints A(8,2) and B(0,6), we can substitute the values into the formula:
Slope of AB = (6 - 2) / (0 - 8)
Slope of AB = 4 / -8
Slope of AB = -1/2
Since the perpendicular bisector has a negative reciprocal slope, the slope of the perpendicular bisector is:
Slope of perpendicular bisector = -1 / (-1/2) = 2
Step 3: Writing the equation using the midpoint and slope.
Now that we have the midpoint (4,4) and the slope (2), we can use the point-slope form of a line equation:
y - y1 = m(x - x1)
Substituting the midpoint (4,4) and the slope (2) into the equation:
y - 4 = 2(x - 4)
Expanding and simplifying the equation:
y - 4 = 2x - 8
y = 2x - 8 + 4
y = 2x - 4
So, the equation of the perpendicular bisector of segment AB is y = 2x - 4.