Find the equation of the perpendicular bisector of the line joining the points (3,8) and (2,-5)?
m=y2-y1/x2-x1
m1=13
since lines are perpendicular m1m2=-1
m2=-1/13
using formula Y-Y1=m(X-X1)
Substite any of the points
Y-8=-1/13(X-3)
open bracket
y-8=-x/13+3/13
13y-104=-x+3
13y+x-104-3=0
13y+x-107=0
To find the equation of the perpendicular bisector of the line joining the points (3,8) and (2,-5), we need to determine two things:
1. The midpoint of the line segment connecting the given points.
2. The slope of the perpendicular bisector.
Let's begin by finding the midpoint. The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) is given by:
Midpoint = ((x₁ + x₂) / 2 , (y₁ + y₂) / 2)
In this case, the two points are (3,8) and (2,-5):
Midpoint = ((3 + 2) / 2 , (8 + (-5)) / 2)
= (5/2 , 3/2)
= (2.5 , 1.5)
Now that we have the midpoint, we can find the slope of the line connecting the two given points using the slope formula:
Slope = (y₂ - y₁) / (x₂ - x₁)
Substituting the values from our given points:
Slope = (-5 - 8) / (2 - 3)
= (-13) / (-1)
= 13
The slope of the line connecting the two given points is 13.
To find the slope of the perpendicular bisector, we need to find the negative reciprocal of the slope of the line connecting the two points.
Negative reciprocal of 13 = -(1/13)
So, the slope of the perpendicular bisector is -(1/13).
Now that we have the midpoint (2.5, 1.5) and the slope -(1/13), we can use the point-slope form to write the equation of the perpendicular bisector:
y - y₁ = m(x - x₁)
Substituting the values:
y - 1.5 = -(1/13)(x - 2.5)
Simplifying the equation, we can multiply -13 on both sides:
-13(y - 1.5) = x - 2.5
-13y + 19.5 = x - 2.5
x + 13y = 22
Therefore, the equation of the perpendicular bisector of the line joining the points (3,8) and (2,-5) is x + 13y = 22.
find the midpoint of the given line
the bisector is perpendicular
... so its slope is the negative-reciprocal of the given line
use point-slope to write the equation
A(3, 8), M(x,y), B(2, -5).
m1 = (-5-8)/(2-3) = -13/-1 = 13.
m2 = -1/13,
x-3 = 2-x,
X = 2.5.
y-8 = -5-y,
Y = 1.5.
(2.5, 1.5),
Y = mx + b,
1.5 = (-1/13)2.5 + b,
b = 1.692.
Eq: Y = (-1/13)x + 1.692.
Why did the line join the points (3,8) and (2,-5) go to therapy?
Because it had issues with its slope and needed to work on its self-intersecting behavior!
To find the equation of the perpendicular bisector, we need to find the midpoint of the line segment connecting the two points and determine the slope of that line. Since the points are (3,8) and (2,-5), the midpoint is:
Midpoint = ((3+2)/2, (8+(-5))/2)
= (2.5, 1.5)
Now, let's find the slope of the line connecting the two points using the formula:
slope = (y2 - y1)/(x2 - x1)
slope = (-5 - 8)/(2 - 3)
= (-13)/(-1)
= 13
Since the perpendicular bisector will have a negative reciprocal slope, the slope of the perpendicular bisector will be -1/13.
We can now use the point-slope form of a line to find the equation of the perpendicular bisector. Using the point (2.5, 1.5) and the slope -1/13, the equation is:
y - 1.5 = (-1/13)(x - 2.5)
Simplifying further, we get:
y - 1.5 = (-1/13)x + 2.5/13
So, the equation of the perpendicular bisector is:
y = (-1/13)x + 2.5/13 + 1.5
Or in standard form:
x + 13y = 10.5