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Find the linear equation, in STANDARD FORM, for the perpendicular bisector of line FG for point F (8, -15) and point G (-4, -5)

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2 Answers

The perpendicular bisector of FG contains the midpoint (because it "bisects"); so, let's find that first
by finding the average of the x-values and the average of the y-values:
 
( (8+-4)/2 , (-15+-5)/2) 
( 4/2 , -20/2 )
( 2, -10 ) is the midpoint
 
The perpendicular bisector must have a perpendicular slope. Recall that the product of perpendicular slopes is -1; so, let's find the slope by using slope formula then find it's opposite (+/-) reciprocal.
 
slope = Δy / Δx = [ y1 - y2 ] / [x1 - x2 ]
= [-15 - (-5) ] / [ 8 - (-4) ]
= ( -15 + 5) / ( 8 + 4)
= -10/12
= -5/6
 
perpendicular slope = +6/5
as -5/6 × 6/5 = -1
 
Now, substitute midpoint (2,-10) & slope 6/5 into point-slope form:
y - y0 = m( x - x0 )
 
y - (-10) = (6/5)( x - 2 )     (distribute on right)
 
y + 10 = (6/5)x - 12/5     (multiply all terms by 5)
 
5y + 50 = 6x - 12   (recall standard form: ax + by = c)
 
-6x + 5y = -62
 
 
The slope of line segment FG is m=  ((-15) - (-5)) / (8 - (-4)) = (-15 + 5)/( 8  + 4) = -10/12 = -5/6
 
The midpoint of line segment FG is m = {( 8 + -4)/2 ,  ( -15 + -5 ) / 2} = ( 4/2 , -20/2 ) = (2, -10) 
 
Any line that is perpendicular MUST have slope that is the negative reciprocal, which in this case is 6/5.
It shall pass through (2, -10)
 
General equation of the line:   y = Mx + b
 
 So the intercept B = y - M*x
 
 We plug in M=6/5, x = 2, y = -10
 
 B = -10 - 6/5*2 = -10 - 12/5 = (-50 - 12)/5 = -62/5 
 
 The equation of the perpendicular bisector is y = 6/5 X - 62/5
 
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There is a way to check this answer. Read on......
 
The intercept of the equation of line FG using poit (-4,-5) is
   B = y - M*x = -5 - (-5/6)*(-4) = -5 - 20/6 = -5 - 10/3 = -15/3 - 10/3 = -25/3
So the equation of line FG is y = (-5/6)X - 25/3
 
TO find where the two lines meet, we set their equations equal and solve:
 
   (-5/6)X - 25/3 = (6/5)X - 62/5
 
 The common denominator of 3,5, and 6 is 30. Multiplying everything by 30 cancels the fractions:
   -25X - 250 = 36X - 372
 
    0 = 36X - 372 + 25X + 250
    0 = 61X - 122
   122 = 61X
   X = 2
 
Plugging this into the original equations:
(-5/6)2 - 25/3 = (6/5)2 - 62/5
 -10/6 - 25/3  = 12/5 - 62/5
  -10/6 - 50/6 = 12/5 - 62/5
  -60/6 = -50/5
  -10  = -10
 
So they intersect at (2, -10) as expected.
They are perpendicular because their slopes are
negative reciprocals of each other.
 
All that is left to show is that (2,-10) is in fact the midpoint of FG.
 
Using the distance formula, the distance from F to the (2,-10) is
   square-root of (   (8 - 2)^2 + (-15 - (-10))^2 )
   square-root of (  6^2  + (-5)^2 )
   square-root of ( 36 + 25 ) = square-root of (61)
    which in slang terms is called "rad61".  
 
"rad" is short for "radical" which means the square root.
 So square root of 61 and "rad61" means the same thing.
 
So if the distance from G to (2,-10) is rad61 , we're done.
 
  square-root of (  (-4 - 2)^2 +  (-5 - (-10))^2 ) =
 square-root of (   (-6)^2 + (-5 + 10)^2 ) =
  square-root of (  36 + (5^2) ) =
  square-root of ( 36 + 25) = square-root of (61) = rad61 
 
The equation of the perpendicular bisector is PROVEN to be y = 6/5 X - 62/5.
There is no more doubt. The answer is 100% correct.
 
In algebra, you must learn to check your answers like this, especially
if there are a lot of fractions and radicals