Final answer:
The perpendicular bisector of the line segment joining the points (0, 6) and (4, -4) does not match any of the answer choices provided. By calculating the midpoint and the slope of the original line, then finding the negative reciprocal of that slope for the perpendicular bisector, we derived the equation y = (2/5)x + 9/5, which is not present among the given options.
Step-by-step explanation:
To determine the equation of the perpendicular bisector of the line segment joining the points (0, 6) and (4, -4), we first need to calculate the midpoint of the segment as well as the slope of the original line. The midpoint (M) can be found by averaging the x-coordinates and y-coordinates of the given points. The slope (m) of the line connecting (0, 6) and (4, -4) is the change in y divided by the change in x.
The midpoint M is found using the formula:
- M = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the coordinates, we get:
- M = ((0 + 4)/2, (6 - 4)/2)
- M = (2, 1)
The slope m of the original line is:
- m = (y2 - y1) / (x2 - x1)
- m = (-4 - 6) / (4 - 0)
- m = -10/4
- m = -5/2
The slope of the perpendicular bisector will be the negative reciprocal of m. So, the slope for the perpendicular bisector is:
Now we use point-slope form to write the equation of the perpendicular bisector:
y - y1 = m(x - x1)
y - 1 = (2/5)(x - 2)
Simplifying this, we get:
y = (2/5)x + (2/5)(2) + 1
y = (2/5)x + 4/5 + 5/5
y = (2/5)x + 9/5
Since none of the answer options match the equation we found, it is possible that the options A, B, C, or D could be a mistake. However, if we multiply both sides of the equation by 5 to clear the fraction, we could see an equation that resembles one of the provided options:
5y = 2x + 9
y = (2/5)x + (9/5)
This makes it clear that none of the given options A, B, C, or D is the correct equation of the perpendicular bisector. They may be a typographical error in the question or the given choices.