Answer:
To find the equation of the perpendicular bisector of the segment with endpoints G(-8, -3) and H(4, -7), we need to determine two things: the midpoint of the segment and the slope of the perpendicular bisector.
1. Midpoint:
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the midpoint formula:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Given G(-8, -3) and H(4, -7), we can find the midpoint M:
M = ((-8 + 4) / 2, (-3 + -7) / 2)
M = (-4 / 2, -10 / 2)
M = (-2, -5)
So the midpoint of the segment GH is M(-2, -5).
2. Slope of the Perpendicular Bisector:
The slope of the segment GH can be found using the slope formula:
Slope = (y₂ - y₁) / (x₂ - x₁)
Given G(-8, -3) and H(4, -7), we can find the slope of GH:
Slope of GH = (-7 - (-3)) / (4 - (-8))
Slope of GH = (-7 + 3) / (4 + 8)
Slope of GH = -4 / 12
Slope of GH = -1/3
The slope of the perpendicular bisector will be the negative reciprocal of the slope of GH. So the slope of the perpendicular bisector will be 3.
Now we have the midpoint (-2, -5) and the slope 3. We can use the point-slope form of a line to write the equation of the perpendicular bisector:
y - y₁ = m(x - x₁)
Using the values: (-2, -5) for (x₁, y₁) and 3 for m, the equation becomes:
y - (-5) = 3(x - (-2))
y + 5 = 3(x + 2)
y + 5 = 3x + 6
y = 3x + 6 - 5
y = 3x + 1
Therefore, the equation of the perpendicular bisector of the segment GH is y = 3x + 1.
Explanation: