What is the equation of the perpendicular bisector of the line between (6, 3) and (3, 4)?
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The equation of the line between the points (6, 3) and (3, 4) is y – 3 = (x – 6)[(4 – 3)/(3 – 6)]
=> y – 3 = (x – 6)*( -1/3)
The slope of the line is (-1/3). The equation of the line perpendicular of this is 3. The midpoint of the given points is [( 6 + 3)/2 , (4 + 3)/2] = ( 9/2 , 7/2 )
The equation of the perpendicular bisector is y – 7/2 = 3( x – 9/2)
=> 2y – 7 = 6x – 27
=> 6x – 2y – 20 = 0
=> 3x – y – 10 = 0
The required equation of the perpendicular bisector is 3x – y – 10 = 0
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Since the line is the bisectrix of the segment whose endpoints are (6, 3) and (3, 4), the midpoint of the segment is on this line.
We'll calculate the midpoint:
x mid = (6+3)/2
x mid = 9/2
y mid = (3+4)/2
y mid = 7/2
Since the bisectrix is perpendicular on the segment, the product of the slopes of the perpendicular lines is -1.
First, we'll write the equation of the segment:
(6-3)/(x-3) = (3-4)/(y-4)
3/(x-3) = -1/(y-4)
-x + 3 = 3y - 12
We'll put the equation in the slope intercept form:
3y = -x + 15
y = -x/3 + 5
The slope of the segment line is m1 = -1/3
m2*m1 = -1
m2 = -1/m1
m2 = 3
The equation of the perpendicular line is:
y - ymid = m2(x - xmid)
y - 7/2 = 3(x - 9/2)
(2y - 7)/2 = (6x - 27)/2
We'll simplify and we'll get:
2y - 7 = 6x - 27
The equation of the perpendicular bisectrix line is 6x - 2y - 20 = 0.
We'll simplify and we'll get:
3x - y - 10 = 0
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