Answer:
To find the equation of the perpendicular bisector of a line segment, we need to determine the midpoint of the line segment and the slope of the line segment.
Given the endpoints (1, 3) and (-7, 7), we can find the midpoint by using the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Midpoint = ((1 + -7) / 2, (3 + 7) / 2)
Midpoint = (-6 / 2, 10 / 2)
Midpoint = (-3, 5)
The midpoint of the line segment is (-3, 5).
Next, we need to find the slope of the line segment. The slope can be calculated using the formula:
Slope = (y2 - y1) / (x2 - x1)
Slope = (7 - 3) / (-7 - 1)
Slope = 4 / -8
Slope = -1/2
The slope of the line segment is -1/2.
The perpendicular bisector of a line segment has a slope that is the negative reciprocal of the slope of the line segment. Therefore, the slope of the perpendicular bisector is 2.
Now, we have the midpoint (-3, 5) and the slope 2. We can use the point-slope form of a linear equation to find the equation of the perpendicular bisector:
y - y1 = m(x - x1)
Plugging in the values, we get:
y - 5 = 2(x - (-3))
y - 5 = 2(x + 3)
y - 5 = 2x + 6
y = 2x + 11
Therefore, the equation of the perpendicular bisector of the line segment with endpoints (1, 3) and (-7, 7) is y = 2x + 11.
I hope this helps! Let me know if you have any other questions.