write the equation of the perpendicular bisector of the line segment whose end points are A(-7,-8) and B(-9,4).
To find the equation of the perpendicular bisector of a line segment, we need to determine two things: the midpoint of the line segment and the slope of the line perpendicular to it.
Step 1: Find the midpoint
The midpoint of a line segment with endpoints A(x1, y1) and B(x2, y2) can be found using the midpoint formula:
midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Given A(-7, -8) and B(-9, 4), we can calculate the midpoint as follows:
midpoint = ((-7 + -9) / 2, (-8 + 4) / 2)
= (-16 / 2, -4 / 2)
= (-8, -2)
So, the midpoint of the line segment AB is (-8, -2).
Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of its slope. To find the slope of the line segment AB, we use the slope formula:
slope = (y2 - y1) / (x2 - x1)
Given A(-7, -8) and B(-9, 4), we can calculate the slope as follows:
slope of AB = (4 - (-8)) / (-9 - (-7))
= 12 / (-2)
= -6
The slope of the perpendicular bisector will be the negative reciprocal of -6, which is 1/6.
Step 3: Write the equation of the perpendicular bisector
Now that we have the midpoint (-8, -2) and the slope 1/6, we can use the point-slope form of a line to write the equation of the perpendicular bisector.
The point-slope form is:
y - y1 = m(x - x1)
Substituting the values we have, we get:
y - (-2) = 1/6(x - (-8))
y + 2 = 1/6(x + 8)
y + 2 = 1/6x + 8/6
To simplify, we can multiply both sides by 6 to get rid of the fraction:
6(y + 2) = x + 8
6y + 12 = x + 8
x - 6y = -4
Therefore, the equation of the perpendicular bisector is x - 6y = -4.
slope of AB = (4+8)/(-9+7) = 12/-2 = -6
so the slope of the perpendicular = 1/6
midpoint of AB = (-8,-2)
equation of our line:
y + 2 = (1/6)(x+8)
6y + 12 = x+8
x - 6y = 4