Write an equation in point slope form for the perpendicular bisector of the segment with endpoints A(-3,-3) and B(5,6)
A(-3,-3), M(x,y), B(5,6).
x = (-3+5) / 2 = 1.
y = (-3+6) / 2 = 1 1/2 = 3/2.
The slope of a straight line is the same at all points on the line.
m = (6-(-3)) / (5-(-3)) = 9/8.
m2 = -8/9 = Negative reciprocal of m.
(1,3/2),(x,y),
m2 = (y-3/2) / (x-1) = -8/9,
Cross multiply:
y-3/2 = -8/9(x-1).
To find the equation of the perpendicular bisector of a line segment, we need to determine the slope of the line containing the segment, and then find the negative reciprocal of that slope. Additionally, we need to find the midpoint of the segment.
First, let's find the slope of the line containing segment AB. The formula for slope between two points (x₁, y₁) and (x₂, y₂) is given by:
m = (y₂ - y₁) / (x₂ - x₁)
In this case, the coordinates of A are (-3, -3) and the coordinates of B are (5, 6). Substituting these values into the slope formula:
m = (6 - (-3)) / (5 - (-3))
= 9 / 8
The slope of the line containing segment AB is 9/8.
Now, let's find the midpoint of the segment AB. The midpoint formula between two points (x₁, y₁) and (x₂, y₂) is given by:
(x, y) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Substituting the coordinates of A and B into the midpoint formula:
(x, y) = ((-3 + 5) / 2, (-3 + 6) / 2)
= (2 / 2, 3 / 2)
= (1, 3/2)
The midpoint of segment AB is (1, 3/2).
Now, we can use the slope of the line containing segment AB and the midpoint to write the equation of the perpendicular bisector in point-slope form.
The point-slope form of a linear equation is: y - y₁ = m(x - x₁)
Substituting the slope (m = -8/9) and the midpoint (x₁ = 1, y₁ = 3/2) into the equation:
y - 3/2 = -8/9(x - 1)
This is the equation in point-slope form for the perpendicular bisector of the segment AB.