Find an equation for the perpendicular bisector of the line segment whose endpoints are (-8,7) and (4,3)
To find the equation of the perpendicular bisector of a line segment, you'll need to follow these steps:
1. Find the midpoint of the line segment.
2. Calculate the slope of the line segment.
3. Determine the negative reciprocal of the slope to find the slope of the perpendicular line.
4. Use the slope and the midpoint to write the equation of the perpendicular bisector in slope-intercept form.
Let's work through these steps:
1. Finding the midpoint:
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the formula:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
For the endpoints (-8, 7) and (4, 3), the midpoint is:
Midpoint = ((-8 + 4) / 2, (7 + 3) / 2)
= (-2, 5)
2. Calculating the slope:
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
Slope = (y₂ - y₁) / (x₂ - x₁)
Using the endpoints (-8, 7) and (4, 3), the slope of the line segment is:
Slope = (3 - 7) / (4 - (-8))
= (-4) / 12
= -1/3
3. Determining the negative reciprocal of the slope:
The negative reciprocal of a slope, m, is -1/m. Since the slope of the line segment is -1/3, the slope of the perpendicular bisector will be the negative reciprocal, which is 3.
4. Writing the equation of the perpendicular bisector:
We have the slope of the perpendicular bisector, 3, and the midpoint, (-2, 5).
To write the equation of a line in slope-intercept form (y = mx + b), we substitute the slope and the coordinates of the midpoint:
y = mx + b
5 = 3(-2) + b
5 = -6 + b
b = 5 + 6
b = 11
Therefore, the equation of the perpendicular bisector is:
y = 3x + 11
To find the equation for the perpendicular bisector of a line segment, we first need to find the midpoint of the line segment whose endpoints are given. The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by:
midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Using the coordinates given, the midpoint is:
midpoint = ((-8 + 4) / 2, (7 + 3) / 2)
= (-4 / 2, 10 / 2)
= (-2, 5)
The slope of the line segment is given by:
slope = (y₂ - y₁) / (x₂ - x₁)
Using the coordinates given, the slope of the line segment is:
slope = (3 - 7) / (4 - (-8))
= (-4) / (12)
= -1/3
Since the perpendicular bisector has a negative reciprocal slope, the slope of the perpendicular bisector is:
perpendicular_slope = -1 / slope
= -1 / (-1/3)
= 3
Now, we have the slope of the perpendicular bisector and the midpoint. Using the point-slope form of the equation of a line, which is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope, we can substitute the values:
y - 5 = 3(x - (-2))
Simplifying, we get:
y - 5 = 3(x + 2)
Finally, expanding the equation, we get the equation of the perpendicular bisector:
y - 5 = 3x + 6
slope of original line = (3-7)/(4+8) = -4/12 = -1/3
so the slope of our line = +3
so
y = 3 x + b
it goes through the point halfway between those two original ones
x = (-8+4)/2 = -2
y = (7+3)/2 = 5
so (-2 , 5)
use that to find b