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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)

Show work ALGEBRAICALLY.
No rounded decimals.

### 2 Answers by Expert Tutors

Kathy M. | ANY MATH -I can break it down to the basics for you!ANY MATH -I can break it down to the bas...
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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

Andy C. | Math/Physics TutorMath/Physics Tutor
4.9 4.9 (19 lesson ratings) (19)
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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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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".

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