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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giorgiana1976 | Student

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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