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Question 1151109: Find The Equation Which Is The Perpendicular Bisector Of The Line Joining (4,7) And (5,3). Please I Need The Full Explaination Please
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Given Points: (x1,y1) = (4,7) and (x2,y2) = (5,3)
Find the slope:
m = (y2 - y1)/(x2 - x1)
m = (3 - 7)/(5 - 4)
m = -4/1
The slope of the line through the two given points is -4/1 or -4. It is handy to keep that "1" around in the denominator for when we flip the fraction in the next part below.
Apply the negative reciprocal.
We flip the fraction going from -4/1 to -1/4
Then we flip the sign from negative to positive: -1/4 turns into 1/4
The perpendicular line will have a slope of 1/4.
Note how the original slope and perpendicular slope multiply to -1
(original slope)*(perpendicular slope) = (-4/1)*(1/4) = -1
This applies to any pair of perpendicular lines as long as neither line is vertical.
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Apply the midpoint formula
Let's apply the notation (Xm, Ym) for the midpoint's location. The 'm' refers to 'midpoint' just so we can tell the x and y values apart.
Add up the x coordinates and divide by 2
Xm = (x1+x2)/2
Xm = (4+5)/2
Xm = 9/2
Xm = 4.5
This is it the x coordinate of the midpoint
Repeat the same basic steps for the y coordinates
Ym = (y1+y2)/2
Ym = (7+3)/2
Ym = 10/2
Ym = 5
This is it the y coordinate of the midpoint
The midpoint is (Xm, Ym) = (4.5, 5)
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The perpendicular bisector will have two properties:
(A) It will have a slope of 1/4 (equivalent to 0.25)
(B) It will go through the point(4.5, 5)
This is because the perpendicular bisector is both perpendicular to the original segment, and it also bisects (aka cuts in half) the original segment.
So we need to find an equation in the form y = mx+b such that
m = 1/4 is the slope
(x,y) = (4.5, 5) is the point this line goes through
Plug those three pieces of info into the equation. Then solve for b
y = mx+b
5 = (1/4)*4.5+b
5 = 0.25*4.5+b
5 = 1.125+b
1.125+b = 5
b+1.125 = 5
b+1.125-1.125 = 5-1.125 ...... subtract 1.125 from both sides
b = 3.875
b = 3+0.875
b = 3+875/1000
b = 3000/1000+875/1000
b = 3875/1000
b = (31*125)/(8*125)
b = 31/8
The equation of the perpendicular bisector, in terms of fractions, is y = (1/4)x + 31/8 (which you can write as  )
In terms of decimal form, the perpendicular bisector is y = 0.25x + 3.875 (this is just another way to express the final answer; both equations highlighted in red are equivalent forms)
Diagram:
The red points A and B are the original points given (4,7) and (5,3). The blue point C is the midpoint (4.5, 5). The purple line is the perpendicular bisector of the green line segment. The angle formed between these two lines is 90 degrees, as shown by the square angle marker.
The green segment AB is cut in half (or bisected) by the purple line. This means AC = CB, or that both smaller pieces are equal to one another.
The diagram was generated by GeoGebra (free graphing software).
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